Practice sheet

Fresh questions from the whole chapter, to do on paper.

New questions Include answers

To save a PDF, choose “Save as PDF” in the print dialog.

Describe the end behavior of $f(x) = 7x^{ 3 }$ .
1. As $x \to \infty$ , $f(x) \to \infty$ ; as $x \to -\infty$ , $f(x) \to -\infty$
2. As $x \to \pm\infty$ , $f(x) \to \infty$
3. As $x \to \infty$ , $f(x) \to -\infty$ ; as $x \to -\infty$ , $f(x) \to \infty$
Describe the end behavior of $f(x) = 3x^{ 7 }$ .
1. As $x \to \infty$ , $f(x) \to \infty$ ; as $x \to -\infty$ , $f(x) \to -\infty$
2. As $x \to \pm\infty$ , $f(x) \to \infty$
3. As $x \to \infty$ , $f(x) \to -\infty$ ; as $x \to -\infty$ , $f(x) \to \infty$
Describe the end behavior of $f(x) = 3x^{ 3 }$ .
1. As $x \to \infty$ , $f(x) \to \infty$ ; as $x \to -\infty$ , $f(x) \to -\infty$
2. As $x \to \pm\infty$ , $f(x) \to \infty$
3. As $x \to \infty$ , $f(x) \to -\infty$ ; as $x \to -\infty$ , $f(x) \to \infty$
Describe the end behavior of $f(x) = 8x^{ 3 }$ .
1. As $x \to \infty$ , $f(x) \to \infty$ ; as $x \to -\infty$ , $f(x) \to -\infty$
2. As $x \to \pm\infty$ , $f(x) \to \infty$
3. As $x \to \infty$ , $f(x) \to -\infty$ ; as $x \to -\infty$ , $f(x) \to \infty$
Describe the end behavior of $f(x) = 4x^{ 5 }$ .
1. As $x \to \infty$ , $f(x) \to \infty$ ; as $x \to -\infty$ , $f(x) \to -\infty$
2. As $x \to \pm\infty$ , $f(x) \to \infty$
3. As $x \to \infty$ , $f(x) \to -\infty$ ; as $x \to -\infty$ , $f(x) \to \infty$
Describe the end behavior of $f(x) = 7x^{ 5 }$ .
1. As $x \to \infty$ , $f(x) \to \infty$ ; as $x \to -\infty$ , $f(x) \to -\infty$
2. As $x \to \pm\infty$ , $f(x) \to \infty$
3. As $x \to \infty$ , $f(x) \to -\infty$ ; as $x \to -\infty$ , $f(x) \to \infty$
Describe the end behavior of $f(x) = 5x^{ 7 }$ .
1. As $x \to \infty$ , $f(x) \to \infty$ ; as $x \to -\infty$ , $f(x) \to -\infty$
2. As $x \to \pm\infty$ , $f(x) \to \infty$
3. As $x \to \infty$ , $f(x) \to -\infty$ ; as $x \to -\infty$ , $f(x) \to \infty$
Describe the end behavior of $f(x) = 5x^{ 3 }$ .
1. As $x \to \infty$ , $f(x) \to \infty$ ; as $x \to -\infty$ , $f(x) \to -\infty$
2. As $x \to \pm\infty$ , $f(x) \to \infty$
3. As $x \to \infty$ , $f(x) \to -\infty$ ; as $x \to -\infty$ , $f(x) \to \infty$
Describe the end behavior of $f(x) = 7x^{ 7 }$ .
1. As $x \to \infty$ , $f(x) \to \infty$ ; as $x \to -\infty$ , $f(x) \to -\infty$
2. As $x \to \pm\infty$ , $f(x) \to \infty$
3. As $x \to \infty$ , $f(x) \to -\infty$ ; as $x \to -\infty$ , $f(x) \to \infty$
Describe the end behavior of $f(x) = 2x^{ 3 }$ .
1. As $x \to \infty$ , $f(x) \to \infty$ ; as $x \to -\infty$ , $f(x) \to -\infty$
2. As $x \to \pm\infty$ , $f(x) \to \infty$
3. As $x \to \infty$ , $f(x) \to -\infty$ ; as $x \to -\infty$ , $f(x) \to \infty$
Describe the end behavior of $f(x) = 5x^{ 5 }$ .
1. As $x \to \infty$ , $f(x) \to \infty$ ; as $x \to -\infty$ , $f(x) \to -\infty$
2. As $x \to \pm\infty$ , $f(x) \to \infty$
3. As $x \to \infty$ , $f(x) \to -\infty$ ; as $x \to -\infty$ , $f(x) \to \infty$
Describe the end behavior of $f(x) = 2x^{ 5 }$ .
1. As $x \to \infty$ , $f(x) \to \infty$ ; as $x \to -\infty$ , $f(x) \to -\infty$
2. As $x \to \pm\infty$ , $f(x) \to \infty$
3. As $x \to \infty$ , $f(x) \to -\infty$ ; as $x \to -\infty$ , $f(x) \to \infty$
Describe the end behavior of $f(x) = 9x^{ 3 }$ .
1. As $x \to \infty$ , $f(x) \to \infty$ ; as $x \to -\infty$ , $f(x) \to -\infty$
2. As $x \to \pm\infty$ , $f(x) \to \infty$
3. As $x \to \infty$ , $f(x) \to -\infty$ ; as $x \to -\infty$ , $f(x) \to \infty$
Describe the end behavior of $f(x) = 4x^{ 3 }$ .
1. As $x \to \infty$ , $f(x) \to \infty$ ; as $x \to -\infty$ , $f(x) \to -\infty$
2. As $x \to \pm\infty$ , $f(x) \to \infty$
3. As $x \to \infty$ , $f(x) \to -\infty$ ; as $x \to -\infty$ , $f(x) \to \infty$
Describe the end behavior of $f(x) = 3x^{ 5 }$ .
1. As $x \to \infty$ , $f(x) \to \infty$ ; as $x \to -\infty$ , $f(x) \to -\infty$
2. As $x \to \pm\infty$ , $f(x) \to \infty$
3. As $x \to \infty$ , $f(x) \to -\infty$ ; as $x \to -\infty$ , $f(x) \to \infty$
Describe the end behavior of $f(x) = 6x^{ 7 }$ .
1. As $x \to \infty$ , $f(x) \to \infty$ ; as $x \to -\infty$ , $f(x) \to -\infty$
2. As $x \to \pm\infty$ , $f(x) \to \infty$
3. As $x \to \infty$ , $f(x) \to -\infty$ ; as $x \to -\infty$ , $f(x) \to \infty$
Describe the end behavior of $f(x) = 8x^{ 7 }$ .
1. As $x \to \infty$ , $f(x) \to \infty$ ; as $x \to -\infty$ , $f(x) \to -\infty$
2. As $x \to \pm\infty$ , $f(x) \to \infty$
3. As $x \to \infty$ , $f(x) \to -\infty$ ; as $x \to -\infty$ , $f(x) \to \infty$
Describe the end behavior of $f(x) = 8x^{ 5 }$ .
1. As $x \to \infty$ , $f(x) \to \infty$ ; as $x \to -\infty$ , $f(x) \to -\infty$
2. As $x \to \pm\infty$ , $f(x) \to \infty$
3. As $x \to \infty$ , $f(x) \to -\infty$ ; as $x \to -\infty$ , $f(x) \to \infty$
Describe the end behavior of $f(x) = 9x^{ 7 }$ .
1. As $x \to \infty$ , $f(x) \to \infty$ ; as $x \to -\infty$ , $f(x) \to -\infty$
2. As $x \to \pm\infty$ , $f(x) \to \infty$
3. As $x \to \infty$ , $f(x) \to -\infty$ ; as $x \to -\infty$ , $f(x) \to \infty$
Describe the end behavior of $f(x) = 4x^{ 7 }$ .
1. As $x \to \infty$ , $f(x) \to \infty$ ; as $x \to -\infty$ , $f(x) \to -\infty$
2. As $x \to \pm\infty$ , $f(x) \to \infty$
3. As $x \to \infty$ , $f(x) \to -\infty$ ; as $x \to -\infty$ , $f(x) \to \infty$
Describe the end behavior of $f(x) = 6x^{ 3 }$ .
1. As $x \to \infty$ , $f(x) \to \infty$ ; as $x \to -\infty$ , $f(x) \to -\infty$
2. As $x \to \pm\infty$ , $f(x) \to \infty$
3. As $x \to \infty$ , $f(x) \to -\infty$ ; as $x \to -\infty$ , $f(x) \to \infty$
Describe the end behavior of $f(x) = 6x^{ 5 }$ .
1. As $x \to \infty$ , $f(x) \to \infty$ ; as $x \to -\infty$ , $f(x) \to -\infty$
2. As $x \to \pm\infty$ , $f(x) \to \infty$
3. As $x \to \infty$ , $f(x) \to -\infty$ ; as $x \to -\infty$ , $f(x) \to \infty$
Describe the end behavior of $f(x) = 2x^{ 7 }$ .
1. As $x \to \infty$ , $f(x) \to \infty$ ; as $x \to -\infty$ , $f(x) \to -\infty$
2. As $x \to \pm\infty$ , $f(x) \to \infty$
3. As $x \to \infty$ , $f(x) \to -\infty$ ; as $x \to -\infty$ , $f(x) \to \infty$
Describe the end behavior of $f(x) = 9x^{ 5 }$ .
1. As $x \to \infty$ , $f(x) \to \infty$ ; as $x \to -\infty$ , $f(x) \to -\infty$
2. As $x \to \pm\infty$ , $f(x) \to \infty$
3. As $x \to \infty$ , $f(x) \to -\infty$ ; as $x \to -\infty$ , $f(x) \to \infty$
Describe the end behavior of $f(x) = -9x^{ 6 }$ .
1. As $x \to \pm\infty$ , $f(x) \to -\infty$
2. As $x \to \pm\infty$ , $f(x) \to \infty$
3. As $x \to \infty$ , $f(x) \to -\infty$ ; as $x \to -\infty$ , $f(x) \to \infty$
Describe the end behavior of $f(x) = -8x^{ 4 }$ .
1. As $x \to \pm\infty$ , $f(x) \to -\infty$
2. As $x \to \pm\infty$ , $f(x) \to \infty$
3. As $x \to \infty$ , $f(x) \to -\infty$ ; as $x \to -\infty$ , $f(x) \to \infty$
Describe the end behavior of $f(x) = -6x^{ 4 }$ .
1. As $x \to \pm\infty$ , $f(x) \to -\infty$
2. As $x \to \pm\infty$ , $f(x) \to \infty$
3. As $x \to \infty$ , $f(x) \to -\infty$ ; as $x \to -\infty$ , $f(x) \to \infty$
Describe the end behavior of $f(x) = -9x^{ 2 }$ .
1. As $x \to \pm\infty$ , $f(x) \to -\infty$
2. As $x \to \pm\infty$ , $f(x) \to \infty$
3. As $x \to \infty$ , $f(x) \to -\infty$ ; as $x \to -\infty$ , $f(x) \to \infty$
Describe the end behavior of $f(x) = -2x^{ 6 }$ .
1. As $x \to \pm\infty$ , $f(x) \to -\infty$
2. As $x \to \pm\infty$ , $f(x) \to \infty$
3. As $x \to \infty$ , $f(x) \to -\infty$ ; as $x \to -\infty$ , $f(x) \to \infty$
Describe the end behavior of $f(x) = -6x^{ 2 }$ .
1. As $x \to \pm\infty$ , $f(x) \to -\infty$
2. As $x \to \pm\infty$ , $f(x) \to \infty$
3. As $x \to \infty$ , $f(x) \to -\infty$ ; as $x \to -\infty$ , $f(x) \to \infty$
Describe the end behavior of $f(x) = -3x^{ 4 }$ .
1. As $x \to \pm\infty$ , $f(x) \to -\infty$
2. As $x \to \pm\infty$ , $f(x) \to \infty$
3. As $x \to \infty$ , $f(x) \to -\infty$ ; as $x \to -\infty$ , $f(x) \to \infty$
Describe the end behavior of $f(x) = -3x^{ 2 }$ .
1. As $x \to \pm\infty$ , $f(x) \to -\infty$
2. As $x \to \pm\infty$ , $f(x) \to \infty$
3. As $x \to \infty$ , $f(x) \to -\infty$ ; as $x \to -\infty$ , $f(x) \to \infty$
Describe the end behavior of $f(x) = -2x^{ 2 }$ .
1. As $x \to \pm\infty$ , $f(x) \to -\infty$
2. As $x \to \pm\infty$ , $f(x) \to \infty$
3. As $x \to \infty$ , $f(x) \to -\infty$ ; as $x \to -\infty$ , $f(x) \to \infty$
Describe the end behavior of $f(x) = -4x^{ 2 }$ .
1. As $x \to \pm\infty$ , $f(x) \to -\infty$
2. As $x \to \pm\infty$ , $f(x) \to \infty$
3. As $x \to \infty$ , $f(x) \to -\infty$ ; as $x \to -\infty$ , $f(x) \to \infty$
Describe the end behavior of $f(x) = -8x^{ 6 }$ .
1. As $x \to \pm\infty$ , $f(x) \to -\infty$
2. As $x \to \pm\infty$ , $f(x) \to \infty$
3. As $x \to \infty$ , $f(x) \to -\infty$ ; as $x \to -\infty$ , $f(x) \to \infty$
Describe the end behavior of $f(x) = -9x^{ 4 }$ .
1. As $x \to \pm\infty$ , $f(x) \to -\infty$
2. As $x \to \pm\infty$ , $f(x) \to \infty$
3. As $x \to \infty$ , $f(x) \to -\infty$ ; as $x \to -\infty$ , $f(x) \to \infty$
Describe the end behavior of $f(x) = -3x^{ 6 }$ .
1. As $x \to \pm\infty$ , $f(x) \to -\infty$
2. As $x \to \pm\infty$ , $f(x) \to \infty$
3. As $x \to \infty$ , $f(x) \to -\infty$ ; as $x \to -\infty$ , $f(x) \to \infty$
Describe the end behavior of $f(x) = -4x^{ 6 }$ .
1. As $x \to \pm\infty$ , $f(x) \to -\infty$
2. As $x \to \pm\infty$ , $f(x) \to \infty$
3. As $x \to \infty$ , $f(x) \to -\infty$ ; as $x \to -\infty$ , $f(x) \to \infty$
Describe the end behavior of $f(x) = -8x^{ 2 }$ .
1. As $x \to \pm\infty$ , $f(x) \to -\infty$
2. As $x \to \pm\infty$ , $f(x) \to \infty$
3. As $x \to \infty$ , $f(x) \to -\infty$ ; as $x \to -\infty$ , $f(x) \to \infty$
Describe the end behavior of $f(x) = -7x^{ 6 }$ .
1. As $x \to \pm\infty$ , $f(x) \to -\infty$
2. As $x \to \pm\infty$ , $f(x) \to \infty$
3. As $x \to \infty$ , $f(x) \to -\infty$ ; as $x \to -\infty$ , $f(x) \to \infty$
Describe the end behavior of $f(x) = -4x^{ 4 }$ .
1. As $x \to \pm\infty$ , $f(x) \to -\infty$
2. As $x \to \pm\infty$ , $f(x) \to \infty$
3. As $x \to \infty$ , $f(x) \to -\infty$ ; as $x \to -\infty$ , $f(x) \to \infty$
Describe the end behavior of $f(x) = -5x^{ 6 }$ .
1. As $x \to \pm\infty$ , $f(x) \to -\infty$
2. As $x \to \pm\infty$ , $f(x) \to \infty$
3. As $x \to \infty$ , $f(x) \to -\infty$ ; as $x \to -\infty$ , $f(x) \to \infty$
Describe the end behavior of $f(x) = -6x^{ 6 }$ .
1. As $x \to \pm\infty$ , $f(x) \to -\infty$
2. As $x \to \pm\infty$ , $f(x) \to \infty$
3. As $x \to \infty$ , $f(x) \to -\infty$ ; as $x \to -\infty$ , $f(x) \to \infty$
Describe the end behavior of $f(x) = -7x^{ 4 }$ .
1. As $x \to \pm\infty$ , $f(x) \to -\infty$
2. As $x \to \pm\infty$ , $f(x) \to \infty$
3. As $x \to \infty$ , $f(x) \to -\infty$ ; as $x \to -\infty$ , $f(x) \to \infty$
Describe the end behavior of $f(x) = -5x^{ 4 }$ .
1. As $x \to \pm\infty$ , $f(x) \to -\infty$
2. As $x \to \pm\infty$ , $f(x) \to \infty$
3. As $x \to \infty$ , $f(x) \to -\infty$ ; as $x \to -\infty$ , $f(x) \to \infty$
Describe the end behavior of $f(x) = -2x^{ 4 }$ .
1. As $x \to \pm\infty$ , $f(x) \to -\infty$
2. As $x \to \pm\infty$ , $f(x) \to \infty$
3. As $x \to \infty$ , $f(x) \to -\infty$ ; as $x \to -\infty$ , $f(x) \to \infty$
Describe the end behavior of $f(x) = -5x^{ 2 }$ .
1. As $x \to \pm\infty$ , $f(x) \to -\infty$
2. As $x \to \pm\infty$ , $f(x) \to \infty$
3. As $x \to \infty$ , $f(x) \to -\infty$ ; as $x \to -\infty$ , $f(x) \to \infty$
Describe the end behavior of $f(x) = -7x^{ 2 }$ .
1. As $x \to \pm\infty$ , $f(x) \to -\infty$
2. As $x \to \pm\infty$ , $f(x) \to \infty$
3. As $x \to \infty$ , $f(x) \to -\infty$ ; as $x \to -\infty$ , $f(x) \to \infty$
$p(x) = 7x^2 - 9x^5 + 9$ . What is the leading coefficient of $p$ ?
$f(x) = 5x^2 - 10x + 20$ . What is the $x$ -coordinate of the vertex?
$f(x) = x^2 + 10x + 21$ . What is the smallest value of $f(x)$ ?
Draw the graph of $y = -(x - h)^2 + 6$ for $h = 0$ : place the vertex and one more point.
Find all the zeros of $p(x) = (x - 4)(x + 2)^2$ .
$p(x) = (x - 4)^2(x + 9)$ . What does the graph of $p$ do at $x = 4$ ?
1. It touches the x-axis and turns back
2. It crosses the x-axis
3. It does not reach the x-axis
$p(x) = (x - 5)^2(x + 1)$ . What does the graph of $p$ do at $x = 5$ ?
1. It touches the x-axis and turns back
2. It crosses the x-axis
3. It does not reach the x-axis
$p(x) = (x - 7)^2(x + 4)$ . What does the graph of $p$ do at $x = 7$ ?
1. It touches the x-axis and turns back
2. It crosses the x-axis
3. It does not reach the x-axis
$p(x) = (x - 4)^2(x + 3)$ . What does the graph of $p$ do at $x = 4$ ?
1. It touches the x-axis and turns back
2. It crosses the x-axis
3. It does not reach the x-axis
$p(x) = (x - 5)^2(x + 2)$ . What does the graph of $p$ do at $x = 5$ ?
1. It touches the x-axis and turns back
2. It crosses the x-axis
3. It does not reach the x-axis
$p(x) = (x - 7)^2(x + 7)$ . What does the graph of $p$ do at $x = 7$ ?
1. It touches the x-axis and turns back
2. It crosses the x-axis
3. It does not reach the x-axis
$p(x) = (x - 2)^2(x + 6)$ . What does the graph of $p$ do at $x = 2$ ?
1. It touches the x-axis and turns back
2. It crosses the x-axis
3. It does not reach the x-axis
$p(x) = (x - 7)^2(x + 8)$ . What does the graph of $p$ do at $x = 7$ ?
1. It touches the x-axis and turns back
2. It crosses the x-axis
3. It does not reach the x-axis
$p(x) = (x - 2)^2(x + 7)$ . What does the graph of $p$ do at $x = 2$ ?
1. It touches the x-axis and turns back
2. It crosses the x-axis
3. It does not reach the x-axis
$p(x) = (x - 8)^2(x + 6)$ . What does the graph of $p$ do at $x = 8$ ?
1. It touches the x-axis and turns back
2. It crosses the x-axis
3. It does not reach the x-axis
$p(x) = (x - 8)^2(x + 1)$ . What does the graph of $p$ do at $x = 8$ ?
1. It touches the x-axis and turns back
2. It crosses the x-axis
3. It does not reach the x-axis
$p(x) = (x - 4)^2(x + 8)$ . What does the graph of $p$ do at $x = 4$ ?
1. It touches the x-axis and turns back
2. It crosses the x-axis
3. It does not reach the x-axis
$p(x) = (x - 6)^2(x + 2)$ . What does the graph of $p$ do at $x = 6$ ?
1. It touches the x-axis and turns back
2. It crosses the x-axis
3. It does not reach the x-axis
$p(x) = (x - 8)^2(x + 2)$ . What does the graph of $p$ do at $x = 8$ ?
1. It touches the x-axis and turns back
2. It crosses the x-axis
3. It does not reach the x-axis
$p(x) = (x - 1)^2(x + 7)$ . What does the graph of $p$ do at $x = 1$ ?
1. It touches the x-axis and turns back
2. It crosses the x-axis
3. It does not reach the x-axis
$p(x) = (x - 5)^2(x + 6)$ . What does the graph of $p$ do at $x = 5$ ?
1. It touches the x-axis and turns back
2. It crosses the x-axis
3. It does not reach the x-axis
$p(x) = (x - 3)^2(x + 9)$ . What does the graph of $p$ do at $x = 3$ ?
1. It touches the x-axis and turns back
2. It crosses the x-axis
3. It does not reach the x-axis
$p(x) = (x - 2)^2(x + 9)$ . What does the graph of $p$ do at $x = 2$ ?
1. It touches the x-axis and turns back
2. It crosses the x-axis
3. It does not reach the x-axis
$p(x) = (x - 7)^2(x + 3)$ . What does the graph of $p$ do at $x = 7$ ?
1. It touches the x-axis and turns back
2. It crosses the x-axis
3. It does not reach the x-axis
$p(x) = (x - 2)^2(x + 5)$ . What does the graph of $p$ do at $x = 2$ ?
1. It touches the x-axis and turns back
2. It crosses the x-axis
3. It does not reach the x-axis
$p(x) = (x - 4)^2(x + 1)$ . What does the graph of $p$ do at $x = 4$ ?
1. It touches the x-axis and turns back
2. It crosses the x-axis
3. It does not reach the x-axis
$p(x) = (x - 8)^2(x + 8)$ . What does the graph of $p$ do at $x = 8$ ?
1. It touches the x-axis and turns back
2. It crosses the x-axis
3. It does not reach the x-axis
$p(x) = (x - 3)^2(x + 8)$ . What does the graph of $p$ do at $x = 3$ ?
1. It touches the x-axis and turns back
2. It crosses the x-axis
3. It does not reach the x-axis
$p(x) = (x - 1)^2(x + 3)$ . What does the graph of $p$ do at $x = 1$ ?
1. It touches the x-axis and turns back
2. It crosses the x-axis
3. It does not reach the x-axis
$p(x) = (x - 1)^2(x + 4)$ . What does the graph of $p$ do at $x = 1$ ?
1. It touches the x-axis and turns back
2. It crosses the x-axis
3. It does not reach the x-axis
$p(x) = (x - 9)^2(x + 3)$ . What does the graph of $p$ do at $x = 9$ ?
1. It touches the x-axis and turns back
2. It crosses the x-axis
3. It does not reach the x-axis
$p(x) = (x - 9)^2(x + 6)$ . What does the graph of $p$ do at $x = 9$ ?
1. It touches the x-axis and turns back
2. It crosses the x-axis
3. It does not reach the x-axis
$p(x) = (x - 9)^2(x + 9)$ . What does the graph of $p$ do at $x = 9$ ?
1. It touches the x-axis and turns back
2. It crosses the x-axis
3. It does not reach the x-axis
$p(x) = (x - 8)^2(x + 4)$ . What does the graph of $p$ do at $x = 8$ ?
1. It touches the x-axis and turns back
2. It crosses the x-axis
3. It does not reach the x-axis
$p(x) = (x - 6)^2(x + 6)$ . What does the graph of $p$ do at $x = 6$ ?
1. It touches the x-axis and turns back
2. It crosses the x-axis
3. It does not reach the x-axis
$p(x) = (x - 7)^2(x + 1)$ . What does the graph of $p$ do at $x = 7$ ?
1. It touches the x-axis and turns back
2. It crosses the x-axis
3. It does not reach the x-axis
$p(x) = (x - 8)^2(x + 5)$ . What does the graph of $p$ do at $x = 8$ ?
1. It touches the x-axis and turns back
2. It crosses the x-axis
3. It does not reach the x-axis
$p(x) = (x - 9)^2(x + 1)$ . What does the graph of $p$ do at $x = 9$ ?
1. It touches the x-axis and turns back
2. It crosses the x-axis
3. It does not reach the x-axis
$p(x) = (x - 6)^2(x + 9)$ . What does the graph of $p$ do at $x = 6$ ?
1. It touches the x-axis and turns back
2. It crosses the x-axis
3. It does not reach the x-axis
$p(x) = (x - 5)^2(x + 3)$ . What does the graph of $p$ do at $x = 5$ ?
1. It touches the x-axis and turns back
2. It crosses the x-axis
3. It does not reach the x-axis
$p(x) = (x - 6)^2(x + 1)$ . What does the graph of $p$ do at $x = 6$ ?
1. It touches the x-axis and turns back
2. It crosses the x-axis
3. It does not reach the x-axis
$p(x) = (x - 2)^2(x + 3)$ . What does the graph of $p$ do at $x = 2$ ?
1. It touches the x-axis and turns back
2. It crosses the x-axis
3. It does not reach the x-axis
$p(x) = (x - 6)^2(x + 4)$ . What does the graph of $p$ do at $x = 6$ ?
1. It touches the x-axis and turns back
2. It crosses the x-axis
3. It does not reach the x-axis
$p(x) = (x - 4)^2(x + 6)$ . What does the graph of $p$ do at $x = 4$ ?
1. It touches the x-axis and turns back
2. It crosses the x-axis
3. It does not reach the x-axis
$p(x) = (x - 9)^2(x + 2)$ . What does the graph of $p$ do at $x = 9$ ?
1. It touches the x-axis and turns back
2. It crosses the x-axis
3. It does not reach the x-axis
$p(x) = (x - 6)^2(x + 7)$ . What does the graph of $p$ do at $x = 6$ ?
1. It touches the x-axis and turns back
2. It crosses the x-axis
3. It does not reach the x-axis
$p(x) = (x - 4)^2(x + 7)$ . What does the graph of $p$ do at $x = 4$ ?
1. It touches the x-axis and turns back
2. It crosses the x-axis
3. It does not reach the x-axis
$p(x) = (x - 9)^2(x + 7)$ . What does the graph of $p$ do at $x = 9$ ?
1. It touches the x-axis and turns back
2. It crosses the x-axis
3. It does not reach the x-axis
$p(x) = (x - 3)^2(x + 4)$ . What does the graph of $p$ do at $x = 3$ ?
1. It touches the x-axis and turns back
2. It crosses the x-axis
3. It does not reach the x-axis
$p(x) = (x - 1)^2(x + 8)$ . What does the graph of $p$ do at $x = 1$ ?
1. It touches the x-axis and turns back
2. It crosses the x-axis
3. It does not reach the x-axis
$p(x) = (x - 4)^2(x + 5)$ . What does the graph of $p$ do at $x = 4$ ?
1. It touches the x-axis and turns back
2. It crosses the x-axis
3. It does not reach the x-axis
$p(x) = (x - 3)^2(x + 1)$ . What does the graph of $p$ do at $x = 3$ ?
1. It touches the x-axis and turns back
2. It crosses the x-axis
3. It does not reach the x-axis
$p(x) = (x - 1)^2(x + 2)$ . What does the graph of $p$ do at $x = 1$ ?
1. It touches the x-axis and turns back
2. It crosses the x-axis
3. It does not reach the x-axis
$p(x) = (x - 4)^2(x + 4)$ . What does the graph of $p$ do at $x = 4$ ?
1. It touches the x-axis and turns back
2. It crosses the x-axis
3. It does not reach the x-axis
$p(x) = (x - 3)^2(x + 5)$ . What does the graph of $p$ do at $x = 3$ ?
1. It touches the x-axis and turns back
2. It crosses the x-axis
3. It does not reach the x-axis
What is the remainder when $p(x) = x^3 - 5x + 7$ is divided by $x - 3$ ?
Is $x - 3$ a factor of $p(x) = x^2 - 7x + 12$ ?
1. Yes, because $p(3) = 0$
2. No, because $p(-3) \ne 0$
3. Only if dividing $p(x)$ by $x - 3$ leaves the remainder 3
Is $x - 5$ a factor of $p(x) = x^2 - 8x + 15$ ?
1. Yes, because $p(5) = 0$
2. No, because $p(-5) \ne 0$
3. Only if dividing $p(x)$ by $x - 5$ leaves the remainder 5
Is $x - 1$ a factor of $p(x) = x^2 - 6x + 5$ ?
1. Yes, because $p(1) = 0$
2. No, because $p(-1) \ne 0$
3. Only if dividing $p(x)$ by $x - 1$ leaves the remainder 1
Is $x - 3$ a factor of $p(x) = x^2 - 9x + 18$ ?
1. Yes, because $p(3) = 0$
2. No, because $p(-3) \ne 0$
3. Only if dividing $p(x)$ by $x - 3$ leaves the remainder 3
Is $x - 6$ a factor of $p(x) = x^2 - 7x + 6$ ?
1. Yes, because $p(6) = 0$
2. No, because $p(-6) \ne 0$
3. Only if dividing $p(x)$ by $x - 6$ leaves the remainder 6
Is $x - 1$ a factor of $p(x) = x^2 - 4x + 3$ ?
1. Yes, because $p(1) = 0$
2. No, because $p(-1) \ne 0$
3. Only if dividing $p(x)$ by $x - 1$ leaves the remainder 1
Is $x - 2$ a factor of $p(x) = x^2 - 5x + 6$ ?
1. Yes, because $p(2) = 0$
2. No, because $p(-2) \ne 0$
3. Only if dividing $p(x)$ by $x - 2$ leaves the remainder 2
Is $x - 6$ a factor of $p(x) = x^2 - 9x + 18$ ?
1. Yes, because $p(6) = 0$
2. No, because $p(-6) \ne 0$
3. Only if dividing $p(x)$ by $x - 6$ leaves the remainder 6
Is $x - 2$ a factor of $p(x) = x^2 - 6x + 8$ ?
1. Yes, because $p(2) = 0$
2. No, because $p(-2) \ne 0$
3. Only if dividing $p(x)$ by $x - 2$ leaves the remainder 2
Is $x - 4$ a factor of $p(x) = x^2 - 6x + 8$ ?
1. Yes, because $p(4) = 0$
2. No, because $p(-4) \ne 0$
3. Only if dividing $p(x)$ by $x - 4$ leaves the remainder 4
Is $x - 5$ a factor of $p(x) = x^2 - 9x + 20$ ?
1. Yes, because $p(5) = 0$
2. No, because $p(-5) \ne 0$
3. Only if dividing $p(x)$ by $x - 5$ leaves the remainder 5
Is $x - 1$ a factor of $p(x) = x^2 - 7x + 6$ ?
1. Yes, because $p(1) = 0$
2. No, because $p(-1) \ne 0$
3. Only if dividing $p(x)$ by $x - 1$ leaves the remainder 1
Is $x - 6$ a factor of $p(x) = x^2 - 10x + 24$ ?
1. Yes, because $p(6) = 0$
2. No, because $p(-6) \ne 0$
3. Only if dividing $p(x)$ by $x - 6$ leaves the remainder 6
Is $x - 3$ a factor of $p(x) = x^2 - 4x + 3$ ?
1. Yes, because $p(3) = 0$
2. No, because $p(-3) \ne 0$
3. Only if dividing $p(x)$ by $x - 3$ leaves the remainder 3
Is $x - 4$ a factor of $p(x) = x^2 - 7x + 12$ ?
1. Yes, because $p(4) = 0$
2. No, because $p(-4) \ne 0$
3. Only if dividing $p(x)$ by $x - 4$ leaves the remainder 4
Is $x - 5$ a factor of $p(x) = x^2 - 6x + 5$ ?
1. Yes, because $p(5) = 0$
2. No, because $p(-5) \ne 0$
3. Only if dividing $p(x)$ by $x - 5$ leaves the remainder 5
Is $x - 1$ a factor of $p(x) = x^2 - 3x + 2$ ?
1. Yes, because $p(1) = 0$
2. No, because $p(-1) \ne 0$
3. Only if dividing $p(x)$ by $x - 1$ leaves the remainder 1
Is $x - 2$ a factor of $p(x) = x^2 - 3x + 2$ ?
1. Yes, because $p(2) = 0$
2. No, because $p(-2) \ne 0$
3. Only if dividing $p(x)$ by $x - 2$ leaves the remainder 2
Is $x - 3$ a factor of $p(x) = x^2 - 5x + 6$ ?
1. Yes, because $p(3) = 0$
2. No, because $p(-3) \ne 0$
3. Only if dividing $p(x)$ by $x - 3$ leaves the remainder 3
Is $x - 2$ a factor of $p(x) = x^2 - 8x + 12$ ?
1. Yes, because $p(2) = 0$
2. No, because $p(-2) \ne 0$
3. Only if dividing $p(x)$ by $x - 2$ leaves the remainder 2
Is $x - 5$ a factor of $p(x) = x^2 - 7x + 10$ ?
1. Yes, because $p(5) = 0$
2. No, because $p(-5) \ne 0$
3. Only if dividing $p(x)$ by $x - 5$ leaves the remainder 5
Is $x - 1$ a factor of $p(x) = x^2 - 5x + 4$ ?
1. Yes, because $p(1) = 0$
2. No, because $p(-1) \ne 0$
3. Only if dividing $p(x)$ by $x - 1$ leaves the remainder 1
Is $x - 4$ a factor of $p(x) = x^2 - 5x + 4$ ?
1. Yes, because $p(4) = 0$
2. No, because $p(-4) \ne 0$
3. Only if dividing $p(x)$ by $x - 4$ leaves the remainder 4
Is $x - 5$ a factor of $p(x) = x^2 - 11x + 30$ ?
1. Yes, because $p(5) = 0$
2. No, because $p(-5) \ne 0$
3. Only if dividing $p(x)$ by $x - 5$ leaves the remainder 5
Is $x - 6$ a factor of $p(x) = x^2 - 11x + 30$ ?
1. Yes, because $p(6) = 0$
2. No, because $p(-6) \ne 0$
3. Only if dividing $p(x)$ by $x - 6$ leaves the remainder 6
Is $x - 2$ a factor of $p(x) = x^2 - 7x + 10$ ?
1. Yes, because $p(2) = 0$
2. No, because $p(-2) \ne 0$
3. Only if dividing $p(x)$ by $x - 2$ leaves the remainder 2
Is $x - 6$ a factor of $p(x) = x^2 - 8x + 12$ ?
1. Yes, because $p(6) = 0$
2. No, because $p(-6) \ne 0$
3. Only if dividing $p(x)$ by $x - 6$ leaves the remainder 6
Is $x - 4$ a factor of $p(x) = x^2 - 10x + 24$ ?
1. Yes, because $p(4) = 0$
2. No, because $p(-4) \ne 0$
3. Only if dividing $p(x)$ by $x - 4$ leaves the remainder 4
Is $x - 4$ a factor of $p(x) = x^2 - 9x + 20$ ?
1. Yes, because $p(4) = 0$
2. No, because $p(-4) \ne 0$
3. Only if dividing $p(x)$ by $x - 4$ leaves the remainder 4
Is $x - 3$ a factor of $p(x) = x^2 - 8x + 15$ ?
1. Yes, because $p(3) = 0$
2. No, because $p(-3) \ne 0$
3. Only if dividing $p(x)$ by $x - 3$ leaves the remainder 3
For which value of $k$ is $x - 2$ a factor of $x^2 + kx + 4$ ?
$x = 4$ is a zero of $p(x) = x^3 - 4x^2 - 1x + 4$ . Find all the zeros of $p$ .
Divide $p(x) = x^3 - 2x^2 - 9x + 18$ by $x - 2$ . What is the quotient?
You know that $2$ is a zero of a cubic $p(x)$ . Put the steps for finding all its zeros in order.
- Check that $p(2) = 0$
- Divide $p(x)$ by $x - 2$
- Solve the quadratic quotient $= 0$
- List $2$ with the solutions of the quadratic
You know that $4$ is a zero of a cubic $p(x)$ . Put the steps for finding all its zeros in order.
- Check that $p(4) = 0$
- Divide $p(x)$ by $x - 4$
- Solve the quadratic quotient $= 0$
- List $4$ with the solutions of the quadratic
You know that $5$ is a zero of a cubic $p(x)$ . Put the steps for finding all its zeros in order.
- Check that $p(5) = 0$
- Divide $p(x)$ by $x - 5$
- Solve the quadratic quotient $= 0$
- List $5$ with the solutions of the quadratic
You know that $3$ is a zero of a cubic $p(x)$ . Put the steps for finding all its zeros in order.
- Check that $p(3) = 0$
- Divide $p(x)$ by $x - 3$
- Solve the quadratic quotient $= 0$
- List $3$ with the solutions of the quadratic
You know that $1$ is a zero of a cubic $p(x)$ . Put the steps for finding all its zeros in order.
- Check that $p(1) = 0$
- Divide $p(x)$ by $x - 1$
- Solve the quadratic quotient $= 0$
- List $1$ with the solutions of the quadratic
You know that $6$ is a zero of a cubic $p(x)$ . Put the steps for finding all its zeros in order.
- Check that $p(6) = 0$
- Divide $p(x)$ by $x - 6$
- Solve the quadratic quotient $= 0$
- List $6$ with the solutions of the quadratic
A polynomial with real coefficients has the zero $4 + 4i$ . Which other number must be a zero?
1. $4 - 4i$
2. $-4 - 4i$
3. $-4 + 4i$
A polynomial with real coefficients has the zero $8 + 7i$ . Which other number must be a zero?
1. $8 - 7i$
2. $-8 - 7i$
3. $-8 + 7i$
A polynomial with real coefficients has the zero $4 + 5i$ . Which other number must be a zero?
1. $4 - 5i$
2. $-4 - 5i$
3. $-4 + 5i$
A polynomial with real coefficients has the zero $6 + 5i$ . Which other number must be a zero?
1. $6 - 5i$
2. $-6 - 5i$
3. $-6 + 5i$
A polynomial with real coefficients has the zero $6 + 2i$ . Which other number must be a zero?
1. $6 - 2i$
2. $-6 - 2i$
3. $-6 + 2i$
A polynomial with real coefficients has the zero $2 + 1i$ . Which other number must be a zero?
1. $2 - 1i$
2. $-2 - 1i$
3. $-2 + 1i$
A polynomial with real coefficients has the zero $8 + 8i$ . Which other number must be a zero?
1. $8 - 8i$
2. $-8 - 8i$
3. $-8 + 8i$
A polynomial with real coefficients has the zero $5 + 4i$ . Which other number must be a zero?
1. $5 - 4i$
2. $-5 - 4i$
3. $-5 + 4i$
A polynomial with real coefficients has the zero $6 + 1i$ . Which other number must be a zero?
1. $6 - 1i$
2. $-6 - 1i$
3. $-6 + 1i$
A polynomial with real coefficients has the zero $4 + 9i$ . Which other number must be a zero?
1. $4 - 9i$
2. $-4 - 9i$
3. $-4 + 9i$
A polynomial with real coefficients has the zero $5 + 9i$ . Which other number must be a zero?
1. $5 - 9i$
2. $-5 - 9i$
3. $-5 + 9i$
A polynomial with real coefficients has the zero $9 + 8i$ . Which other number must be a zero?
1. $9 - 8i$
2. $-9 - 8i$
3. $-9 + 8i$
A polynomial with real coefficients has the zero $2 + 9i$ . Which other number must be a zero?
1. $2 - 9i$
2. $-2 - 9i$
3. $-2 + 9i$
A polynomial with real coefficients has the zero $3 + 9i$ . Which other number must be a zero?
1. $3 - 9i$
2. $-3 - 9i$
3. $-3 + 9i$
A polynomial with real coefficients has the zero $5 + 6i$ . Which other number must be a zero?
1. $5 - 6i$
2. $-5 - 6i$
3. $-5 + 6i$
A polynomial with real coefficients has the zero $6 + 4i$ . Which other number must be a zero?
1. $6 - 4i$
2. $-6 - 4i$
3. $-6 + 4i$
A polynomial with real coefficients has the zero $4 + 6i$ . Which other number must be a zero?
1. $4 - 6i$
2. $-4 - 6i$
3. $-4 + 6i$
A polynomial with real coefficients has the zero $1 + 5i$ . Which other number must be a zero?
1. $1 - 5i$
2. $-1 - 5i$
3. $-1 + 5i$
A polynomial with real coefficients has the zero $6 + 7i$ . Which other number must be a zero?
1. $6 - 7i$
2. $-6 - 7i$
3. $-6 + 7i$
A polynomial with real coefficients has the zero $1 + 2i$ . Which other number must be a zero?
1. $1 - 2i$
2. $-1 - 2i$
3. $-1 + 2i$
A polynomial with real coefficients has the zero $2 + 3i$ . Which other number must be a zero?
1. $2 - 3i$
2. $-2 - 3i$
3. $-2 + 3i$
A polynomial with real coefficients has the zero $9 + 5i$ . Which other number must be a zero?
1. $9 - 5i$
2. $-9 - 5i$
3. $-9 + 5i$
A polynomial with real coefficients has the zero $2 + 4i$ . Which other number must be a zero?
1. $2 - 4i$
2. $-2 - 4i$
3. $-2 + 4i$
A polynomial with real coefficients has the zero $9 + 6i$ . Which other number must be a zero?
1. $9 - 6i$
2. $-9 - 6i$
3. $-9 + 6i$
A polynomial with real coefficients has the zero $7 + 6i$ . Which other number must be a zero?
1. $7 - 6i$
2. $-7 - 6i$
3. $-7 + 6i$
A polynomial with real coefficients has the zero $5 + 2i$ . Which other number must be a zero?
1. $5 - 2i$
2. $-5 - 2i$
3. $-5 + 2i$
A polynomial with real coefficients has the zero $5 + 8i$ . Which other number must be a zero?
1. $5 - 8i$
2. $-5 - 8i$
3. $-5 + 8i$
A polynomial with real coefficients has the zero $8 + 3i$ . Which other number must be a zero?
1. $8 - 3i$
2. $-8 - 3i$
3. $-8 + 3i$
A polynomial with real coefficients has the zero $4 + 8i$ . Which other number must be a zero?
1. $4 - 8i$
2. $-4 - 8i$
3. $-4 + 8i$
A polynomial with real coefficients has the zero $2 + 5i$ . Which other number must be a zero?
1. $2 - 5i$
2. $-2 - 5i$
3. $-2 + 5i$
A polynomial with real coefficients has the zero $1 + 9i$ . Which other number must be a zero?
1. $1 - 9i$
2. $-1 - 9i$
3. $-1 + 9i$
A polynomial with real coefficients has the zero $9 + 3i$ . Which other number must be a zero?
1. $9 - 3i$
2. $-9 - 3i$
3. $-9 + 3i$
A polynomial with real coefficients has the zero $3 + 1i$ . Which other number must be a zero?
1. $3 - 1i$
2. $-3 - 1i$
3. $-3 + 1i$
A polynomial with real coefficients has the zero $2 + 7i$ . Which other number must be a zero?
1. $2 - 7i$
2. $-2 - 7i$
3. $-2 + 7i$
A polynomial with real coefficients has the zero $6 + 9i$ . Which other number must be a zero?
1. $6 - 9i$
2. $-6 - 9i$
3. $-6 + 9i$
A polynomial with real coefficients has the zero $1 + 1i$ . Which other number must be a zero?
1. $1 - 1i$
2. $-1 - 1i$
3. $-1 + 1i$
A polynomial with real coefficients has the zero $9 + 2i$ . Which other number must be a zero?
1. $9 - 2i$
2. $-9 - 2i$
3. $-9 + 2i$
A polynomial with real coefficients has the zero $2 + 2i$ . Which other number must be a zero?
1. $2 - 2i$
2. $-2 - 2i$
3. $-2 + 2i$
A polynomial with real coefficients has the zero $3 + 7i$ . Which other number must be a zero?
1. $3 - 7i$
2. $-3 - 7i$
3. $-3 + 7i$
A polynomial with real coefficients has the zero $8 + 2i$ . Which other number must be a zero?
1. $8 - 2i$
2. $-8 - 2i$
3. $-8 + 2i$
A polynomial with real coefficients has the zero $7 + 4i$ . Which other number must be a zero?
1. $7 - 4i$
2. $-7 - 4i$
3. $-7 + 4i$
A polynomial with real coefficients has the zero $9 + 1i$ . Which other number must be a zero?
1. $9 - 1i$
2. $-9 - 1i$
3. $-9 + 1i$
A polynomial with real coefficients has the zero $8 + 4i$ . Which other number must be a zero?
1. $8 - 4i$
2. $-8 - 4i$
3. $-8 + 4i$
A polynomial with real coefficients has the zero $6 + 6i$ . Which other number must be a zero?
1. $6 - 6i$
2. $-6 - 6i$
3. $-6 + 6i$
A polynomial with real coefficients has the zero $8 + 9i$ . Which other number must be a zero?
1. $8 - 9i$
2. $-8 - 9i$
3. $-8 + 9i$
A polynomial with real coefficients has the zero $4 + 2i$ . Which other number must be a zero?
1. $4 - 2i$
2. $-4 - 2i$
3. $-4 + 2i$
A polynomial with real coefficients has the zero $9 + 7i$ . Which other number must be a zero?
1. $9 - 7i$
2. $-9 - 7i$
3. $-9 + 7i$
A polynomial with real coefficients has the zero $3 + 2i$ . Which other number must be a zero?
1. $3 - 2i$
2. $-3 - 2i$
3. $-3 + 2i$
A polynomial with real coefficients has the zero $7 + 2i$ . Which other number must be a zero?
1. $7 - 2i$
2. $-7 - 2i$
3. $-7 + 2i$
A polynomial with real coefficients has the zero $3 + 4i$ . Which other number must be a zero?
1. $3 - 4i$
2. $-3 - 4i$
3. $-3 + 4i$
How many real zeros does $x^2 + 5x + 10$ have?
1. Two
2. One
3. None: two complex zeros
How many real zeros does $x^2 + 4x + 9$ have?
1. Two
2. One
3. None: two complex zeros
How many real zeros does $x^2 + 2x + 20$ have?
1. Two
2. One
3. None: two complex zeros
How many real zeros does $x^2 + 3x + 6$ have?
1. Two
2. One
3. None: two complex zeros
How many real zeros does $x^2 + 1x + 1$ have?
1. Two
2. One
3. None: two complex zeros
How many real zeros does $x^2 + 5x + 12$ have?
1. Two
2. One
3. None: two complex zeros
How many real zeros does $x^2 + 6x + 14$ have?
1. Two
2. One
3. None: two complex zeros
How many real zeros does $x^2 + 3x + 8$ have?
1. Two
2. One
3. None: two complex zeros
How many real zeros does $x^2 + 4x + 13$ have?
1. Two
2. One
3. None: two complex zeros
How many real zeros does $x^2 + 6x + 12$ have?
1. Two
2. One
3. None: two complex zeros
How many real zeros does $x^2 + 3x + 15$ have?
1. Two
2. One
3. None: two complex zeros
How many real zeros does $x^2 + 5x + 7$ have?
1. Two
2. One
3. None: two complex zeros
How many real zeros does $x^2 + 6x + 15$ have?
1. Two
2. One
3. None: two complex zeros
How many real zeros does $x^2 + 5x + 8$ have?
1. Two
2. One
3. None: two complex zeros
How many real zeros does $x^2 + 3x + 18$ have?
1. Two
2. One
3. None: two complex zeros
How many real zeros does $x^2 + 1x + 4$ have?
1. Two
2. One
3. None: two complex zeros
How many real zeros does $x^2 + 2x + 10$ have?
1. Two
2. One
3. None: two complex zeros
How many real zeros does $x^2 + 3x + 9$ have?
1. Two
2. One
3. None: two complex zeros
How many real zeros does $x^2 + 1x + 10$ have?
1. Two
2. One
3. None: two complex zeros
How many real zeros does $x^2 + 2x + 18$ have?
1. Two
2. One
3. None: two complex zeros
How many real zeros does $x^2 + 2x + 9$ have?
1. Two
2. One
3. None: two complex zeros
How many real zeros does $x^2 + 4x + 19$ have?
1. Two
2. One
3. None: two complex zeros
How many real zeros does $x^2 + 2x + 16$ have?
1. Two
2. One
3. None: two complex zeros
How many real zeros does $x^2 + 6x + 20$ have?
1. Two
2. One
3. None: two complex zeros
How many real zeros does $x^2 + 3x + 20$ have?
1. Two
2. One
3. None: two complex zeros
How many real zeros does $x^2 + 2x + 8$ have?
1. Two
2. One
3. None: two complex zeros
How many real zeros does $x^2 + 4x + 17$ have?
1. Two
2. One
3. None: two complex zeros
How many real zeros does $x^2 + 2x + 15$ have?
1. Two
2. One
3. None: two complex zeros
How many real zeros does $x^2 + 3x + 14$ have?
1. Two
2. One
3. None: two complex zeros
How many real zeros does $x^2 + 1x + 18$ have?
1. Two
2. One
3. None: two complex zeros
How many real zeros does $x^2 + 4x + 15$ have?
1. Two
2. One
3. None: two complex zeros
How many real zeros does $x^2 + 1x + 3$ have?
1. Two
2. One
3. None: two complex zeros
How many real zeros does $x^2 + 2x + 12$ have?
1. Two
2. One
3. None: two complex zeros
How many real zeros does $x^2 + 3x + 19$ have?
1. Two
2. One
3. None: two complex zeros
How many real zeros does $x^2 + 3x + 4$ have?
1. Two
2. One
3. None: two complex zeros
How many real zeros does $x^2 + 3x + 13$ have?
1. Two
2. One
3. None: two complex zeros
How many real zeros does $x^2 + 5x + 16$ have?
1. Two
2. One
3. None: two complex zeros
How many real zeros does $x^2 + 4x + 20$ have?
1. Two
2. One
3. None: two complex zeros
How many real zeros does $x^2 + 4x + 10$ have?
1. Two
2. One
3. None: two complex zeros
How many real zeros does $x^2 + 1x + 16$ have?
1. Two
2. One
3. None: two complex zeros
How many real zeros does $x^2 + 5x + 13$ have?
1. Two
2. One
3. None: two complex zeros
How many real zeros does $x^2 + 6x + 11$ have?
1. Two
2. One
3. None: two complex zeros
How many real zeros does $x^2 + 1x + 8$ have?
1. Two
2. One
3. None: two complex zeros
How many real zeros does $x^2 + 3x + 3$ have?
1. Two
2. One
3. None: two complex zeros
How many real zeros does $x^2 + 4x + 5$ have?
1. Two
2. One
3. None: two complex zeros
How many real zeros does $x^2 + 1x + 13$ have?
1. Two
2. One
3. None: two complex zeros
How many real zeros does $x^2 + 6x + 10$ have?
1. Two
2. One
3. None: two complex zeros
How many real zeros does $x^2 + 1x + 20$ have?
1. Two
2. One
3. None: two complex zeros
How many real zeros does $x^2 + 3x + 5$ have?
1. Two
2. One
3. None: two complex zeros
How many real zeros does $x^2 + 1x + 12$ have?
1. Two
2. One
3. None: two complex zeros
Solve $x^2 - 8x + 25 = 0$ . Type each solution as $a + bi$ .
Add: $(9 + 8i) + (2 + 1i)$ .
Multiply: $(2 + 8i)(2 - 8i)$ .
Expand $(3 + 6i)^2$ .
$x^2 + 36 = 0$ . Enter the solution with positive imaginary part.
$x^2 - 4x + 20 = 0$ . Enter the solution with positive imaginary part.
Find the discriminant of $1x^2 + 7x + 4 = 0$ .
Simplify (x² − 4) ÷ (x − 2), that is, the fraction with numerator x² − 4 and denominator x − 2. Which answer is right?
1. x + 2, for x ≠ 2
2. x − 2
3. x + 2, for every x
Simplify (x² − 25) ÷ (x − 5), that is, the fraction with numerator x² − 25 and denominator x − 5. Which answer is right?
1. x + 5, for x ≠ 5
2. x − 5
3. x + 5, for every x
Simplify (x² − 9) ÷ (x − 3), that is, the fraction with numerator x² − 9 and denominator x − 3. Which answer is right?
1. x + 3, for x ≠ 3
2. x − 3
3. x + 3, for every x
Simplify (x² − 16) ÷ (x − 4), that is, the fraction with numerator x² − 16 and denominator x − 4. Which answer is right?
1. x + 4, for x ≠ 4
2. x − 4
3. x + 4, for every x
Simplify (x² − 36) ÷ (x − 6), that is, the fraction with numerator x² − 36 and denominator x − 6. Which answer is right?
1. x + 6, for x ≠ 6
2. x − 6
3. x + 6, for every x
Simplify (x² − 81) ÷ (x − 9), that is, the fraction with numerator x² − 81 and denominator x − 9. Which answer is right?
1. x + 9, for x ≠ 9
2. x − 9
3. x + 9, for every x
Simplify (x² − 64) ÷ (x − 8), that is, the fraction with numerator x² − 64 and denominator x − 8. Which answer is right?
1. x + 8, for x ≠ 8
2. x − 8
3. x + 8, for every x
Simplify (x² − 49) ÷ (x − 7), that is, the fraction with numerator x² − 49 and denominator x − 7. Which answer is right?
1. x + 7, for x ≠ 7
2. x − 7
3. x + 7, for every x
Multiply and simplify: (x + 8)/(x − 6) · (x − 6)/(x + 9).
Divide and simplify: x/(x + 3) ÷ x²/(x + 3).
Write 1/x + 1/(x + 8) as a single fraction.
Simplify 6/(x − 1) − 9/(x − 1).
What is the least common denominator of 1/(x² − 16) and 1/(x + 4)?
$f(x) = \dfrac{ 4x + 6 }{x - 6}$ has a vertical asymptote $x = c$ . What is $c$ ?
$f(x) = \dfrac{ 5x + 6 }{ 2x + 4 }$ has a horizontal asymptote $y = k$ . What is $k$ ?
What is the horizontal asymptote of $f(x) = \dfrac{ 4x + 1}{x^2 + 6}$ ?
1. $y = 0$
2. $y = 4$
3. There is none, because the degrees are different
What is the horizontal asymptote of $f(x) = \dfrac{ 8x + 1}{x^2 + 5}$ ?
1. $y = 0$
2. $y = 8$
3. There is none, because the degrees are different
What is the horizontal asymptote of $f(x) = \dfrac{ 7x + 1}{x^2 + 6}$ ?
1. $y = 0$
2. $y = 7$
3. There is none, because the degrees are different
What is the horizontal asymptote of $f(x) = \dfrac{ 8x + 1}{x^2 + 6}$ ?
1. $y = 0$
2. $y = 8$
3. There is none, because the degrees are different
What is the horizontal asymptote of $f(x) = \dfrac{ 8x + 1}{x^2 + 1}$ ?
1. $y = 0$
2. $y = 8$
3. There is none, because the degrees are different
What is the horizontal asymptote of $f(x) = \dfrac{ 5x + 1}{x^2 + 4}$ ?
1. $y = 0$
2. $y = 5$
3. There is none, because the degrees are different
What is the horizontal asymptote of $f(x) = \dfrac{ 6x + 1}{x^2 + 2}$ ?
1. $y = 0$
2. $y = 6$
3. There is none, because the degrees are different
What is the horizontal asymptote of $f(x) = \dfrac{ 8x + 1}{x^2 + 3}$ ?
1. $y = 0$
2. $y = 8$
3. There is none, because the degrees are different
What is the horizontal asymptote of $f(x) = \dfrac{ 3x + 1}{x^2 + 1}$ ?
1. $y = 0$
2. $y = 3$
3. There is none, because the degrees are different
What is the horizontal asymptote of $f(x) = \dfrac{ 6x + 1}{x^2 + 6}$ ?
1. $y = 0$
2. $y = 6$
3. There is none, because the degrees are different
What is the horizontal asymptote of $f(x) = \dfrac{ 5x + 1}{x^2 + 6}$ ?
1. $y = 0$
2. $y = 5$
3. There is none, because the degrees are different
What is the horizontal asymptote of $f(x) = \dfrac{ 4x + 1}{x^2 + 4}$ ?
1. $y = 0$
2. $y = 4$
3. There is none, because the degrees are different
What is the horizontal asymptote of $f(x) = \dfrac{ 9x + 1}{x^2 + 9}$ ?
1. $y = 0$
2. $y = 9$
3. There is none, because the degrees are different
What is the horizontal asymptote of $f(x) = \dfrac{ 3x + 1}{x^2 + 6}$ ?
1. $y = 0$
2. $y = 3$
3. There is none, because the degrees are different
What is the horizontal asymptote of $f(x) = \dfrac{ 6x + 1}{x^2 + 3}$ ?
1. $y = 0$
2. $y = 6$
3. There is none, because the degrees are different
What is the horizontal asymptote of $f(x) = \dfrac{ 7x + 1}{x^2 + 9}$ ?
1. $y = 0$
2. $y = 7$
3. There is none, because the degrees are different
What is the horizontal asymptote of $f(x) = \dfrac{ 3x + 1}{x^2 + 4}$ ?
1. $y = 0$
2. $y = 3$
3. There is none, because the degrees are different
What is the horizontal asymptote of $f(x) = \dfrac{ 4x + 1}{x^2 + 2}$ ?
1. $y = 0$
2. $y = 4$
3. There is none, because the degrees are different
What is the horizontal asymptote of $f(x) = \dfrac{ 9x + 1}{x^2 + 4}$ ?
1. $y = 0$
2. $y = 9$
3. There is none, because the degrees are different
What is the horizontal asymptote of $f(x) = \dfrac{ 3x + 1}{x^2 + 8}$ ?
1. $y = 0$
2. $y = 3$
3. There is none, because the degrees are different
What is the horizontal asymptote of $f(x) = \dfrac{ 7x + 1}{x^2 + 7}$ ?
1. $y = 0$
2. $y = 7$
3. There is none, because the degrees are different
What is the horizontal asymptote of $f(x) = \dfrac{ 5x + 1}{x^2 + 9}$ ?
1. $y = 0$
2. $y = 5$
3. There is none, because the degrees are different
What is the horizontal asymptote of $f(x) = \dfrac{ 2x + 1}{x^2 + 6}$ ?
1. $y = 0$
2. $y = 2$
3. There is none, because the degrees are different
What is the horizontal asymptote of $f(x) = \dfrac{ 6x + 1}{x^2 + 8}$ ?
1. $y = 0$
2. $y = 6$
3. There is none, because the degrees are different
What is the horizontal asymptote of $f(x) = \dfrac{ 9x + 1}{x^2 + 1}$ ?
1. $y = 0$
2. $y = 9$
3. There is none, because the degrees are different
What is the horizontal asymptote of $f(x) = \dfrac{ 9x + 1}{x^2 + 6}$ ?
1. $y = 0$
2. $y = 9$
3. There is none, because the degrees are different
What is the horizontal asymptote of $f(x) = \dfrac{ 2x + 1}{x^2 + 3}$ ?
1. $y = 0$
2. $y = 2$
3. There is none, because the degrees are different
What is the horizontal asymptote of $f(x) = \dfrac{ 2x + 1}{x^2 + 7}$ ?
1. $y = 0$
2. $y = 2$
3. There is none, because the degrees are different
What is the horizontal asymptote of $f(x) = \dfrac{ 4x + 1}{x^2 + 1}$ ?
1. $y = 0$
2. $y = 4$
3. There is none, because the degrees are different
What is the horizontal asymptote of $f(x) = \dfrac{ 5x + 1}{x^2 + 1}$ ?
1. $y = 0$
2. $y = 5$
3. There is none, because the degrees are different
What is the horizontal asymptote of $f(x) = \dfrac{ 3x + 1}{x^2 + 2}$ ?
1. $y = 0$
2. $y = 3$
3. There is none, because the degrees are different
What is the horizontal asymptote of $f(x) = \dfrac{ 3x + 1}{x^2 + 3}$ ?
1. $y = 0$
2. $y = 3$
3. There is none, because the degrees are different
What is the horizontal asymptote of $f(x) = \dfrac{ 2x + 1}{x^2 + 9}$ ?
1. $y = 0$
2. $y = 2$
3. There is none, because the degrees are different
What is the horizontal asymptote of $f(x) = \dfrac{ 2x + 1}{x^2 + 1}$ ?
1. $y = 0$
2. $y = 2$
3. There is none, because the degrees are different
What is the horizontal asymptote of $f(x) = \dfrac{ 9x + 1}{x^2 + 2}$ ?
1. $y = 0$
2. $y = 9$
3. There is none, because the degrees are different
What is the horizontal asymptote of $f(x) = \dfrac{ 6x + 1}{x^2 + 7}$ ?
1. $y = 0$
2. $y = 6$
3. There is none, because the degrees are different
What is the horizontal asymptote of $f(x) = \dfrac{ 7x + 1}{x^2 + 4}$ ?
1. $y = 0$
2. $y = 7$
3. There is none, because the degrees are different
What is the horizontal asymptote of $f(x) = \dfrac{ 6x + 1}{x^2 + 4}$ ?
1. $y = 0$
2. $y = 6$
3. There is none, because the degrees are different
What is the horizontal asymptote of $f(x) = \dfrac{ 5x + 1}{x^2 + 8}$ ?
1. $y = 0$
2. $y = 5$
3. There is none, because the degrees are different
What is the horizontal asymptote of $f(x) = \dfrac{ 8x + 1}{x^2 + 7}$ ?
1. $y = 0$
2. $y = 8$
3. There is none, because the degrees are different
What is the horizontal asymptote of $f(x) = \dfrac{ 7x + 1}{x^2 + 5}$ ?
1. $y = 0$
2. $y = 7$
3. There is none, because the degrees are different
What is the horizontal asymptote of $f(x) = \dfrac{ 6x + 1}{x^2 + 1}$ ?
1. $y = 0$
2. $y = 6$
3. There is none, because the degrees are different
What is the horizontal asymptote of $f(x) = \dfrac{ 8x + 1}{x^2 + 8}$ ?
1. $y = 0$
2. $y = 8$
3. There is none, because the degrees are different
What is the horizontal asymptote of $f(x) = \dfrac{ 7x + 1}{x^2 + 8}$ ?
1. $y = 0$
2. $y = 7$
3. There is none, because the degrees are different
What is the horizontal asymptote of $f(x) = \dfrac{ 2x + 1}{x^2 + 8}$ ?
1. $y = 0$
2. $y = 2$
3. There is none, because the degrees are different
What is the horizontal asymptote of $f(x) = \dfrac{ 3x + 1}{x^2 + 5}$ ?
1. $y = 0$
2. $y = 3$
3. There is none, because the degrees are different
What is the horizontal asymptote of $f(x) = \dfrac{ 9x + 1}{x^2 + 7}$ ?
1. $y = 0$
2. $y = 9$
3. There is none, because the degrees are different
What is the horizontal asymptote of $f(x) = \dfrac{ 9x + 1}{x^2 + 3}$ ?
1. $y = 0$
2. $y = 9$
3. There is none, because the degrees are different
What is the horizontal asymptote of $f(x) = \dfrac{ 9x + 1}{x^2 + 5}$ ?
1. $y = 0$
2. $y = 9$
3. There is none, because the degrees are different
What is the horizontal asymptote of $f(x) = \dfrac{ 7x + 1}{x^2 + 2}$ ?
1. $y = 0$
2. $y = 7$
3. There is none, because the degrees are different
Where does the graph of $f(x) = \dfrac{x - 8}{x + 4}$ cross the $x$ -axis? Give $x$ .
$f(x) = \dfrac{(x - 5)(x + 6)}{x - 5}$ . At which $x$ does the graph have a hole?
$f(x) = \dfrac{(x - 1)(x + 3)}{x - 1}$ has a hole at $x = 1$ . What is the $y$ -coordinate of the hole?

Answers

As $x \to \infty$ , $f(x) \to \infty$ ; as $x \to -\infty$ , $f(x) \to -\infty$
As $x \to \pm\infty$ , $f(x) \to -\infty$
$-9$
$1$
$-4$
$y = 6 - x^{2}$
$4,\ -2$
It touches the x-axis and turns back
$19$
Yes, because $p(3) = 0$
$-4$
$4,\ 1,\ -1$
$x^{2} - 9$
1. Check that $p(2) = 0$ 2. Divide $p(x)$ by $x - 2$ 3. Solve the quadratic quotient $= 0$ 4. List $2$ with the solutions of the quadratic
$4 - 4i$
None: two complex zeros
$4 + 3 i,\ 4 - 3 i$
$11 + 9 i$
$68$
$-27 + 36 i$
$6 i$
$2 + 4 i$
$33$
x + 2, for x ≠ 2
$\frac{x + 8}{x + 9}$
$\frac{1}{x}$
$\frac{2 x + 8}{x \left(x + 8\right)}$
$- \frac{3}{x - 1}$
$x^{2} - 16$
$6$
$\frac{5}{2}$
$y = 0$
$8$
$5$
$4$