Print the chapter

Every lesson of the chapter, ready for paper. Exercise sets are lettered A, B, C… and answered at the end.

Include answers

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

1. Power functions and end behavior

Warm-up

Three quick questions from earlier lessons.

Discover

Move the slider and watch the graph of $y = x^n$ for different powers $n$ .

The graph of y = x to the power n. For even n both ends rise; for odd n the left end falls and the right end rises.

For $n = 3$ , what happens on the far left of the graph?

It goes down
It goes up, like $x^2$

Key idea

For large $|x|$ , a polynomial behaves like its leading term, the term with the highest power.

For

a x^n

: if

n

is even, both ends go the same way (up if

a > 0

, down if

a < 0

). If

n

is odd, the ends go opposite ways (the right end up if

a > 0

, down if

a < 0

Worked example

Describe the end behavior of $p(x) = 3x^2 - 2x^5 + 1$ .

Leading term:

-2x^5

The highest power is x⁵, wherever it is written.

Odd degree

5 is odd: the ends go opposite ways.

x \to \infty

p(x) \to -\infty

; as

x \to -\infty

p(x) \to \infty

The coefficient −2 is negative: the right end goes down.

Your turn, guided

Your turn: $q(x) = 4x - 7x^4$ .

Leading term:

-7x^4

The highest power.

Leading coefficient:

-7

, so both ends go down

Type the leading coefficient.

Practice ★

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 $x \to -\infty$ , $f(x) \to \infty$

Watch out!

Practice ★★

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 $x \to -\infty$ , $f(x) \to \infty$

$p(x) = 7x^2 - 9x^5 + 9$ . What is the leading coefficient of $p$ ?
$p(x) = 9x^2 - 7x^5 + 2$ . What is the leading coefficient of $p$ ?
$p(x) = 3x^2 - 5x^5 + 8$ . What is the leading coefficient of $p$ ?

Exit ticket

Three questions to check you have got it.

$p(x) = 7x^2 - 9x^5 + 9$ . What is the leading coefficient of $p$ ?
$p(x) = 9x^2 - 7x^5 + 2$ . What is the leading coefficient of $p$ ?
$p(x) = 3x^2 - 5x^5 + 8$ . What is the leading coefficient of $p$ ?

Summary card

End behavior

Only the leading term (highest power) matters for large $|x|$ .
Even degree: both ends the same way. Odd degree: opposite ways.
A negative leading coefficient flips the picture.

Watch out: the leading term is not always written first.

2. Quadratic functions: vertex and extreme value

Warm-up

Three quick questions from earlier lessons.

Discover

A ball's height is $h(t) = -5t^2 + 20t$ meters after $t$ seconds. Its graph is a parabola that opens downward. When does the ball reach its highest point?

After 2 seconds
After 4 seconds

Key idea

Every quadratic $f(x) = ax^2 + bx + c$ has a vertex, its highest or lowest point.

The vertex is at

x = -\dfrac{b}{2a}

. Its height

f\left(-\dfrac{b}{2a}\right)

is the smallest value of

f

a > 0

, the largest if

a < 0

Completing the square gives the vertex form $f(x) = a(x - h)^2 + k$ , with vertex $(h, k)$ .

Worked example

Find the vertex of $f(x) = 2x^2 - 12x + 5$ .

x = -\dfrac{-12}{2 \cdot 2} = 3

a = 2, b = −12.

f(3) = 18 - 36 + 5 = -13

Evaluate f at the vertex.

Vertex

(3, -13)

, minimum value

-13

a = 2 > 0, so this is the lowest point.

Your turn, guided

Your turn: $f(x) = x^2 + 6x + 11$ .

x = -\dfrac{6}{2} = -3

a = 1, b = 6.

Minimum value:

f(-3) = 2

f(−3) = 9 − 18 + 11.

Practice ★

$f(x) = 5x^2 - 10x + 20$ . What is the $x$ -coordinate of the vertex?
$f(x) = 5x^2 - 40x + 10$ . What is the $x$ -coordinate of the vertex?
$f(x) = 2x^2 - 20x + 19$ . What is the $x$ -coordinate of the vertex?
$f(x) = 5x^2 - 30x + 5$ . What is the $x$ -coordinate of the vertex?

Watch out!

Practice ★★

$f(x) = x^2 + 10x + 21$ . What is the smallest value of $f(x)$ ?
$f(x) = x^2 + 4x + 22$ . What is the smallest value of $f(x)$ ?
$f(x) = x^2 + 10x + 19$ . 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.
Draw the graph of $y = -(x - h)^2 + 2$ for $h = 2$ : place the vertex and one more point.

Exit ticket

Three questions to check you have got it.

$f(x) = x^2 + 10x + 21$ . What is the smallest value of $f(x)$ ?
$f(x) = x^2 + 4x + 22$ . What is the smallest value of $f(x)$ ?
$f(x) = x^2 + 10x + 19$ . What is the smallest value of $f(x)$ ?

Summary card

Quadratic functions

Vertex at $x = -\dfrac{b}{2a}$ .
Its height $f\left(-\dfrac{b}{2a}\right)$ is the minimum ( $a > 0$ ) or maximum ( $a < 0$ ).
Vertex form: $a(x - h)^2 + k$ , vertex $(h, k)$ .

Watch out: the constant term $c$ is $f(0)$ , not the extreme value.

3. Zeros of polynomials and their multiplicity

Warm-up

Three quick questions from earlier lessons.

Discover

$p(x) = (x - 2)(x + 3)$ . For which $x$ is $p(x) = 0$ ?

$x = 2$ and $x = -3$
$x = 2$ and $x = 3$

Key idea

A product is zero when one of its factors is zero. That gives the zeros of a factored polynomial.

The factor

(x - r)^m

gives the zero

r

with multiplicity

m

. If

m

is odd, the graph crosses the

x

-axis at

r

; if

m

is even, it touches the axis and turns back.

Worked example

Describe the zeros of $p(x) = (x - 1)^2(x + 4)$ .

x = 1

, multiplicity 2: the graph touches the axis

x − 1 = 0, and the factor is squared.

x = -4

, multiplicity 1: the graph crosses

x + 4 = 0, the factor appears once.

Your turn, guided

Your turn: $p(x) = (x + 5)^3(x - 2)$ .

x = 2

, multiplicity 1

x − 2 = 0, once: the graph crosses.

x = -5

, multiplicity 3: the graph crosses

x + 5 = 0: type the zero.

Practice ★

Find all the zeros of $p(x) = (x - 4)(x + 2)^2$ .
Find all the zeros of $p(x) = (x - 5)(x + 1)^2$ .
Find all the zeros of $p(x) = (x - 9)(x + 1)^2$ .
Find all the zeros of $p(x) = (x - 3)(x + 1)^2$ .

Watch out!

Practice ★★

$p(x) = (x - 4)^2(x + 9)$ . 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

Which numbers are zeros of $p(x) = x(x - 9)(x + 7)$ ? Tick all of them.
1. $0$
2. $9$
3. $-7$
4. $-9$
5. $7$
Which numbers are zeros of $p(x) = x(x - 2)(x + 4)$ ? Tick all of them.
1. $0$
2. $2$
3. $-4$
4. $-2$
5. $4$

Exit ticket

Three questions to check you have got it.

Find all the zeros of $p(x) = (x - 4)(x + 2)^2$ .
Find all the zeros of $p(x) = (x - 5)(x + 1)^2$ .
Find all the zeros of $p(x) = (x - 9)(x + 1)^2$ .

Summary card

Zeros and multiplicity

$(x - r)$ gives the zero $r$ ; $(x + q)$ gives the zero $-q$ .
Odd multiplicity: the graph crosses. Even: it touches and turns back.

Watch out: check the sign: $x + 3 = 0$ means $x = -3$ .

4. The remainder theorem and the factor theorem

Warm-up

Three quick questions from earlier lessons.

Discover

$p(x) = x^2 - 5x + 6$ . Work out $p(2)$ . What does the answer tell you?

$p(2) = 0$ , so $x - 2$ is a factor
$p(2) = 0$ , so $x + 2$ is a factor

Key idea

Dividing by $x - r$ and evaluating at $r$ are linked.

Remainder theorem: the remainder of

p(x) \div (x - r)

p(r)

. Factor theorem:

x - r

is a factor of

p(x)

exactly when

p(r) = 0

Worked example

Find the remainder of $p(x) = x^3 - 4x + 1$ divided by $x - 2$ .

Remainder

= p(2)

Remainder theorem: evaluate p at 2, not at −2.

p(2) = 1

8 − 8 + 1.

x - 2

is not a factor

The remainder is not 0.

Your turn, guided

Your turn: is $x - 3$ a factor of $p(x) = x^3 - 27$ ?

p(3) = 27 - 27

Factor theorem: compute p(3).

p(3) = 0

, so yes

Type the value of p(3).

Practice ★

What is the remainder when $p(x) = x^3 - 5x + 7$ is divided by $x - 3$ ?
What is the remainder when $p(x) = x^3 - 2x + 6$ is divided by $x - 3$ ?
What is the remainder when $p(x) = x^3 - 7x + 3$ is divided by $x - 2$ ?
What is the remainder when $p(x) = x^3 - 1x + 6$ is divided by $x - 4$ ?

Watch out!

Practice ★★

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

For which value of $k$ is $x - 2$ a factor of $x^2 + kx + 4$ ?
For which value of $k$ is $x - 1$ a factor of $x^2 + kx + 2$ ?

Exit ticket

Three questions to check you have got it.

What is the remainder when $p(x) = x^3 - 5x + 7$ is divided by $x - 3$ ?
What is the remainder when $p(x) = x^3 - 2x + 6$ is divided by $x - 3$ ?
What is the remainder when $p(x) = x^3 - 7x + 3$ is divided by $x - 2$ ?

Summary card

Remainder and factor theorems

Remainder of $p(x) \div (x - r)$ : $p(r)$ .
$x - r$ is a factor $\iff p(r) = 0$ .
For $x + r$ , evaluate at $-r$ .

Watch out: a factor needs a remainder of $0$ .

5. Find all the real zeros from one known zero

Warm-up

Three quick questions from earlier lessons.

Discover

$p(x) = x^3 - 2x^2 - 9x + 18$ and $p(2) = 0$ . How many zeros can a polynomial of degree 3 have?

At most 3
Only 1, the one we know

Key idea

One known zero makes a cubic much easier.

p(r) = 0

, divide

p(x)

x - r

. The quotient

q(x)

has degree one less, and

p(x) = (x - r)\,q(x)

. The other zeros are the zeros of

q

Worked example

Find all the zeros of $p(x) = x^3 - 2x^2 - 9x + 18$ , knowing that $p(2) = 0$ .

p(x) = x^2(x - 2) - 9(x - 2)

Group terms to see the factor x − 2 (or use long or synthetic division).

p(x) = (x - 2)(x^2 - 9)

Factor out x − 2.

Zeros:

2

3

and

-3

x² = 9 has two solutions.

Your turn, guided

Your turn: $p(x) = (x - 1)(x^2 - 16)$ .

x = 1

From the first factor.

x = 4

and

x = -4

x² = 16: type the negative solution.

Practice ★

Divide $p(x) = x^3 - 2x^2 - 9x + 18$ by $x - 2$ . What is the quotient?
Divide $p(x) = x^3 - 6x^2 - 1x + 6$ by $x - 6$ . What is the quotient?
Divide $p(x) = x^3 - 3x^2 - 16x + 48$ by $x - 3$ . What is the quotient?

Watch out!

Practice ★★

$x = 4$ is a zero of $p(x) = x^3 - 4x^2 - 1x + 4$ . Find all the zeros of $p$ .
$x = 3$ is a zero of $p(x) = x^3 - 3x^2 - 16x + 48$ . Find all the zeros of $p$ .
$x = 2$ is a zero of $p(x) = x^3 - 2x^2 - 1x + 2$ . Find all the zeros of $p$ .

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

Exit ticket

Three questions to check you have got it.

$x = 4$ is a zero of $p(x) = x^3 - 4x^2 - 1x + 4$ . Find all the zeros of $p$ .
$x = 3$ is a zero of $p(x) = x^3 - 3x^2 - 16x + 48$ . Find all the zeros of $p$ .
$x = 2$ is a zero of $p(x) = x^3 - 2x^2 - 1x + 2$ . Find all the zeros of $p$ .

Summary card

All zeros from one zero

Check $p(r) = 0$ .
Divide by $x - r$ : $p(x) = (x - r)\,q(x)$ .
Solve $q(x) = 0$ ; list $r$ with those solutions.

Watch out: $x^2 = 9$ gives $3$ and $-3$ .

6. Complex zeros and conjugate pairs

Warm-up

Three quick questions from earlier lessons.

Discover

Solve $x^2 = -9$ . Is there a real number whose square is $-9$ ?

No real number; the solutions are $3i$ and $-3i$
Yes: $3$ and $-3$

Key idea

The number $i$ has $i^2 = -1$ . Complex numbers $a + bi$ give every quadratic its zeros.

For

ax^2 + bx + c

: if

b^2 - 4ac < 0

, there are no real zeros but two complex zeros

\dfrac{-b \pm i\sqrt{4ac - b^2}}{2a}

. A polynomial with real coefficients has complex zeros in conjugate pairs:

p + qi

and

p - qi

Worked example

Solve $x^2 - 4x + 13 = 0$ .

b^2 - 4ac = 16 - 52 = -36

The discriminant decides.

x = \dfrac{4 \pm 6i}{2}

√(−36) = 6i.

x = 2 + 3i

x = 2 - 3i

Divide both parts by 2.

Your turn, guided

Your turn: $x^2 - 2x + 5 = 0$ .

b^2 - 4ac = -16

4 − 20.

x = 1 + 2i

x = 1 - 2i

√(−16) = 4i, and (2 + 4i) ÷ 2 = 1 + 2i. Type this first solution.

Practice ★

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

Watch out!

Practice ★★

Solve $x^2 - 2x + 5 = 0$ . Its solutions are $u + vi$ and $u - vi$ , with $v > 0$ .

Solve $x^2 - 8x + 25 = 0$ . Type each solution as $a + bi$ .
Solve $x^2 - 8x + 32 = 0$ . Type each solution as $a + bi$ .

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. $-4 + 5i$

Exit ticket

Three questions to check you have got it.

Solve $x^2 - 8x + 25 = 0$ . Type each solution as $a + bi$ .
Solve $x^2 - 8x + 32 = 0$ . Type each solution as $a + bi$ .
Solve $x^2 - 4x + 5 = 0$ . Type each solution as $a + bi$ .

Summary card

Complex zeros

$i^2 = -1$ , so $\sqrt{-k^2} = ki$ .
$b^2 - 4ac < 0$ : no real zeros, two complex ones.
Real coefficients: $p + qi$ and $p - qi$ come together.

Watch out: the conjugate changes only the sign of the $i$ part.

7. Adding, subtracting and multiplying complex numbers

Warm-up

Three quick questions from earlier lessons.

Discover

$(2 + 3i) + (4 + 5i)$ . Treat $i$ like a variable. What do you get?

$6 + 8i$
$14$

Key idea

Calculate as with polynomials in $i$ , then use $i^2 = -1$ .

(a + bi) \pm (c + di) = (a \pm c) + (b \pm d)i

(a + bi)(c + di) = (ac - bd) + (ad + bc)i

, since

bd\,i^2 = -bd

. The conjugate of

a + bi

a - bi

, and

(a + bi)(a - bi) = a^2 + b^2

Worked example

$(2 + 3i)(1 - 4i)$ .

2 - 8i + 3i - 12i^2

Distribute.

2 - 5i + 12

i² = −1.

14 - 5i

Combine.

Your turn, guided

Your turn: $(3 + 2i)(3 - 2i)$ .

13

9 + 4.

Practice ★

Add: $(9 + 8i) + (2 + 1i)$ .
Add: $(3 + 4i) + (9 + 2i)$ .

Multiply: $(2 + 8i)(2 - 8i)$ .
Multiply: $(7 + 2i)(7 - 2i)$ .

Watch out!

Practice ★★

Expand $(3 + 6i)^2$ .
Expand $(1 + 6i)^2$ .

Exit ticket

Three questions to check you have got it.

Multiply: $(2 + 8i)(2 - 8i)$ .
Multiply: $(7 + 2i)(7 - 2i)$ .
Multiply: $(8 + 4i)(8 - 4i)$ .

Summary card

Complex arithmetic

Add parts separately; multiply out and use $i^2 = -1$ ; conjugates give a real product.

Watch out: $i^2$ is $-1$ , not $1$ .

8. Quadratic equations with complex solutions

Warm-up

Three quick questions from earlier lessons.

Discover

$x^2 + 9 = 0$ has no real solution. With $i$ , which numbers solve it?

$x = 3i$ and $x = -3i$
$x = 3$ and $x = -3$

Key idea

Every quadratic has solutions among the complex numbers.

x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}

. If the discriminant

b^2 - 4ac < 0

, the solutions are two non-real complex conjugates

p \pm qi

; if

= 0

, one real solution; if

> 0

, two real solutions.

Worked example

$x^2 - 4x + 13 = 0$ .

Discriminant

= -36

16 − 52.

x = \dfrac{4 \pm 6i}{2}

√(−36) = 6i.

x = 2 \pm 3i

Divide both parts by 2.

Your turn, guided

Your turn: the discriminant of $x^2 + 2x + 5 = 0$ .

-16

: two non-real solutions

4 − 20.

Practice ★

$x^2 + 36 = 0$ . Enter the solution with positive imaginary part.
$x^2 + 81 = 0$ . Enter the solution with positive imaginary part.

Find the discriminant of $1x^2 + 7x + 4 = 0$ .

Watch out!

Practice ★★

$x^2 - 4x + 20 = 0$ . Enter the solution with positive imaginary part.
$x^2 - 6x + 34 = 0$ . Enter the solution with positive imaginary part.

Boss problem. Solve $x^2 - 10x + 29 = 0$ with the quadratic formula.

Exit ticket

Three questions to check you have got it.

$x^2 + 36 = 0$ . Enter the solution with positive imaginary part.
$x^2 + 81 = 0$ . Enter the solution with positive imaginary part.
$x^2 + 25 = 0$ . Enter the solution with positive imaginary part.

Summary card

Complex solutions

Discriminant < 0 → $x = p \pm qi$ (conjugates).

Watch out: divide the whole numerator by $2a$ .

9. Simplifying, multiplying and dividing rational expressions

Warm-up

Three quick questions from earlier lessons.

Discover

Simplify 6/9.

2/3
6/9 cannot be simplified

Key idea

Rational expressions work like fractions.

Factor numerators and denominators, then cancel common factors (never terms). Multiply numerators and denominators; to divide, multiply by the reciprocal. Keep the values that make an original denominator zero excluded.

Worked example

Simplify (x² − 9)/(x² + 3x).

(x − 3)(x + 3) / (x(x + 3))

Factor.

(x − 3)/x, for x ≠ 0 and x ≠ −3

Cancel x + 3.

Your turn, guided

Your turn: the value of (x² − 4)/(x − 2), simplified, at x = 3.

5

x + 2 at x = 3.

Practice ★

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

Multiply and simplify: (x + 8)/(x − 6) · (x − 6)/(x + 9).

Watch out!

Practice ★★

Divide and simplify: x/(x + 3) ÷ x²/(x + 3).
Divide and simplify: x/(x + 6) ÷ x²/(x + 6).

Multiply and simplify: (x + 8)/(x − 6) · (x − 6)/(x + 9).

Exit ticket

Three questions to check you have got it.

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

Summary card

Multiplying and dividing rational expressions

Factor, cancel common factors, multiply; divide by multiplying by the reciprocal.

Watch out: cancel factors, never terms.

10. Adding and subtracting rational expressions

Warm-up

Three quick questions from earlier lessons.

Discover

What is 1/2 + 1/3?

Key idea

Add rational expressions over a common denominator.

Factor the denominators, find the least common denominator, rewrite each fraction over it, then add or subtract the numerators only. Simplify at the end.

Worked example

Write 2/x + 3/(x + 1) as one fraction.

x(x + 1)

LCD.

2(x + 1)/(x(x + 1)) + 3x/(x(x + 1))

Rewrite.

(5x + 2)/(x(x + 1))

Add.

Your turn, guided

Your turn: the value of 1/x + 1/x at x = 4.

\tfrac12

2/x at x = 4.

Practice ★

Simplify 6/(x − 1) − 9/(x − 1).
Simplify 9/(x − 1) − 2/(x − 1).

What is the least common denominator of 1/(x² − 16) and 1/(x + 4)?

Watch out!

Practice ★★

Write 1/x + 1/(x + 8) as a single fraction.
Write 1/x + 1/(x + 1) as a single fraction.

Simplify 6/(x − 1) − 9/(x − 1).

Exit ticket

Three questions to check you have got it.

Write 1/x + 1/(x + 8) as a single fraction.
Write 1/x + 1/(x + 1) as a single fraction.
Write 1/x + 1/(x + 4) as a single fraction.

Summary card

Adding rational expressions

Common denominator, then add the numerators.

Watch out: never add the denominators.

11. Rational functions: vertical and horizontal asymptotes

Warm-up

Three quick questions from earlier lessons.

Discover

Move the sliders and watch the graph of $y = \dfrac{a}{x - h}$ .

The graph of y = a over (x − h): two branches separated by a vertical asymptote at x = h, approaching the x-axis far left and far right.

Where is the vertical line that the graph never touches?

At $x = h$
At $x = a$

Key idea

A rational function is a quotient of polynomials.

Vertical asymptotes are where the denominator is 0 (and the numerator is not). Horizontal asymptote: if the top degree is smaller,

y = 0

; if the degrees are equal,

y = \dfrac{\text{leading coefficient on top}}{\text{leading coefficient below}}

; if the top degree is larger, there is none.

Worked example

Find the asymptotes of $f(x) = \dfrac{6x + 1}{2x - 8}$ .

Vertical asymptote:

x = 4

Denominator 0: 2x − 8 = 0.

Horizontal asymptote:

y = \dfrac{6}{2} = 3

Same degree: ratio of leading coefficients.

Your turn, guided

Your turn: $f(x) = \dfrac{5x - 2}{x + 3}$ .

Vertical asymptote:

x = -3

x + 3 = 0.

Horizontal asymptote:

y = 5

Same degree: 5 over 1.

Practice ★

$f(x) = \dfrac{ 4x + 6 }{x - 6}$ has a vertical asymptote $x = c$ . What is $c$ ?
$f(x) = \dfrac{ 6x + 4 }{x - 5}$ has a vertical asymptote $x = c$ . What is $c$ ?
$f(x) = \dfrac{ 2x + 5 }{x - 2}$ has a vertical asymptote $x = c$ . What is $c$ ?
$f(x) = \dfrac{ 5x + 7 }{x - 4}$ has a vertical asymptote $x = c$ . What is $c$ ?

Watch out!

Practice ★★

$f(x) = \dfrac{ 5x + 6 }{ 2x + 4 }$ has a horizontal asymptote $y = k$ . What is $k$ ?
$f(x) = \dfrac{ 2x + 2 }{ 4x + 7 }$ has a horizontal asymptote $y = k$ . What is $k$ ?
$f(x) = \dfrac{ 8x + 5 }{ 4x + 5 }$ 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

Exit ticket

Three questions to check you have got it.

$f(x) = \dfrac{ 5x + 6 }{ 2x + 4 }$ has a horizontal asymptote $y = k$ . What is $k$ ?
$f(x) = \dfrac{ 2x + 2 }{ 4x + 7 }$ has a horizontal asymptote $y = k$ . What is $k$ ?
$f(x) = \dfrac{ 8x + 5 }{ 4x + 5 }$ has a horizontal asymptote $y = k$ . What is $k$ ?

Summary card

Asymptotes of rational functions

Vertical: zeros of the denominator.
Horizontal: compare degrees. Top smaller: $y = 0$ . Equal: ratio of leading coefficients. Top larger: none.

Watch out: use the leading coefficients, not the constants.

12. Rational functions: intercepts and holes

Warm-up

Three quick questions from earlier lessons.

Discover

$f(x) = \dfrac{(x - 2)(x + 1)}{x - 2}$ . Simplified, it looks like $x + 1$ . Is $f$ exactly the same function as $x + 1$ ?

Almost: $f$ is not defined at $x = 2$
Yes, exactly the same

Key idea

The numerator and a cancelled factor tell you where the graph meets the axis and where it has a gap.

x-intercepts: zeros of the numerator that are not zeros of the denominator. Hole: a factor that cancels from top and bottom; the hole is at that

x

, and its height is the value of the simplified rule there. y-intercept:

f(0)

Worked example

Analyze $f(x) = \dfrac{(x - 3)(x + 4)}{(x - 3)(x - 1)}$ .

f(x) = \dfrac{x + 4}{x - 1}

for

x \ne 3

x − 3 cancels.

Hole at

\left(3, \dfrac{7}{2}\right)

The simplified rule at x = 3: 7/2.

x

-intercept:

x = -4

Numerator zero that remains.

Vertical asymptote:

x = 1

Denominator zero that remains.

Your turn, guided

Your turn: $f(x) = \dfrac{(x + 2)(x - 5)}{x + 2}$ .

Hole at

x = -2

x + 2 cancels: f(x) = x − 5 for x ≠ −2.

The hole is at

(-2, -7)

The simplified rule at x = −2: −2 − 5.

Practice ★

Where does the graph of $f(x) = \dfrac{x - 8}{x + 4}$ cross the $x$ -axis? Give $x$ .
Where does the graph of $f(x) = \dfrac{x - 5}{x + 1}$ cross the $x$ -axis? Give $x$ .
Where does the graph of $f(x) = \dfrac{x - 8}{x + 8}$ cross the $x$ -axis? Give $x$ .
Where does the graph of $f(x) = \dfrac{x - 3}{x + 7}$ cross the $x$ -axis? Give $x$ .

Watch out!

Practice ★★

$f(x) = \dfrac{(x - 5)(x + 6)}{x - 5}$ . At which $x$ does the graph have a hole?
$f(x) = \dfrac{(x - 6)(x + 4)}{x - 6}$ . At which $x$ does the graph have a hole?
$f(x) = \dfrac{(x - 2)(x + 4)}{x - 2}$ . 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?
$f(x) = \dfrac{(x - 7)(x + 7)}{x - 7}$ has a hole at $x = 7$ . What is the $y$ -coordinate of the hole?

Exit ticket

Three questions to check you have got it.

$f(x) = \dfrac{(x - 1)(x + 3)}{x - 1}$ has a hole at $x = 1$ . What is the $y$ -coordinate of the hole?
$f(x) = \dfrac{(x - 7)(x + 7)}{x - 7}$ has a hole at $x = 7$ . What is the $y$ -coordinate of the hole?
$f(x) = \dfrac{(x - 9)(x + 1)}{x - 9}$ has a hole at $x = 9$ . What is the $y$ -coordinate of the hole?

Summary card

Intercepts and holes

Cancel common factors: each one leaves a hole.
Hole height: the simplified rule at that $x$ .
$x$ -intercepts: remaining zeros of the numerator; $y$ -intercept: $f(0)$ .

Watch out: zeros of the denominator give asymptotes, not intercepts.

Answers: Polynomial and rational functions

1. Power functions and end behavior

As $x \to \infty$ , $f(x) \to \infty$ ; as $x \to -\infty$ , $f(x) \to -\infty$
As $x \to \infty$ , $f(x) \to \infty$ ; as $x \to -\infty$ , $f(x) \to -\infty$
As $x \to \infty$ , $f(x) \to \infty$ ; as $x \to -\infty$ , $f(x) \to -\infty$
As $x \to \infty$ , $f(x) \to \infty$ ; as $x \to -\infty$ , $f(x) \to -\infty$

As $x \to \pm\infty$ , $f(x) \to -\infty$
As $x \to \pm\infty$ , $f(x) \to -\infty$
As $x \to \pm\infty$ , $f(x) \to -\infty$

$-9$
$-7$
$-5$

$-9$
$-7$
$-5$

2. Quadratic functions: vertex and extreme value

$-4$
$18$
$-6$

$y = 6 - x^{2}$
$y = 2 - \left(x - 2\right)^{2}$

$-4$
$18$
$-6$

3. Zeros of polynomials and their multiplicity

$4,\ -2$
$5,\ -1$
$9,\ -1$
$3,\ -1$

It touches the x-axis and turns back
It touches the x-axis and turns back
It touches the x-axis and turns back

$0$ · $9$ · $-7$
$0$ · $2$ · $-4$

$4,\ -2$
$5,\ -1$
$9,\ -1$

4. The remainder theorem and the factor theorem

$19$
$27$
$-3$
$66$

Yes, because $p(3) = 0$
Yes, because $p(5) = 0$
Yes, because $p(1) = 0$

$-4$
$-3$

$19$
$27$
$-3$

5. Find all the real zeros from one known zero

$x^{2} - 9$
$x^{2} - 1$
$x^{2} - 16$

$4,\ 1,\ -1$
$3,\ 4,\ -4$
$2,\ 1,\ -1$

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

$4,\ 1,\ -1$
$3,\ 4,\ -4$
$2,\ 1,\ -1$

6. Complex zeros and conjugate pairs

None: two complex zeros
None: two complex zeros
None: two complex zeros

$4 + 3 i,\ 4 - 3 i$
$4 + 4 i,\ 4 - 4 i$

$4 - 4i$
$8 - 7i$
$4 - 5i$

$4 + 3 i,\ 4 - 3 i$
$4 + 4 i,\ 4 - 4 i$
$2 + i,\ 2 - i$

7. Adding, subtracting and multiplying complex numbers

$11 + 9 i$
$12 + 6 i$

$68$
$53$

$-27 + 36 i$
$-35 + 12 i$

$68$
$53$
$80$

8. Quadratic equations with complex solutions

$6 i$
$9 i$

$33$

$2 + 4 i$
$3 + 5 i$

$5 + 2 i$

$6 i$
$9 i$
$5 i$

9. Simplifying, multiplying and dividing rational expressions

x + 2, for x ≠ 2
x + 5, for x ≠ 5

$\frac{x + 8}{x + 9}$

$\frac{1}{x}$
$\frac{1}{x}$

$\frac{x + 8}{x + 9}$

x + 2, for x ≠ 2
x + 5, for x ≠ 5
x + 3, for x ≠ 3

10. Adding and subtracting rational expressions

$- \frac{3}{x - 1}$
$\frac{7}{x - 1}$

$x^{2} - 16$

$\frac{2 x + 8}{x \left(x + 8\right)}$
$\frac{2 x + 1}{x \left(x + 1\right)}$

$- \frac{3}{x - 1}$

$\frac{2 x + 8}{x \left(x + 8\right)}$
$\frac{2 x + 1}{x \left(x + 1\right)}$
$\frac{2 x + 4}{x \left(x + 4\right)}$

11. Rational functions: vertical and horizontal asymptotes

$\frac{5}{2}$
$\frac{1}{2}$
$2$

$y = 0$
$y = 0$

$\frac{5}{2}$
$\frac{1}{2}$
$2$

12. Rational functions: intercepts and holes

$4$
$14$

$4$
$14$
$10$