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. Adding, subtracting and multiplying polynomials

Warm-up

Three quick questions from earlier lessons.

Discover

A box made from a 20 cm by 30 cm sheet, with squares of side $x$ cut from the corners, has volume $x(20 - 2x)(30 - 2x)$ . Multiplied out, what is the highest power of $x$ ?

$x^3$
$x^2$

Key idea

Polynomials combine like numbers.

To add or subtract, combine like terms (same power of

x

). To multiply, multiply every term of one by every term of the other, then combine like terms. The degree of a product is the sum of the degrees.

Worked example

$(x + 3)(x^2 - 2x + 5)$ .

x^3 - 2x^2 + 5x

x times each term.

3x^2 - 6x + 15

3 times each term.

x^3 + x^2 - x + 15

Combine like terms.

Your turn, guided

Your turn: the coefficient of $x$ in $(x + 4)(x + 5)$ .

9

5x + 4x.

Practice ★

Expand $(x + 6)(x + 9)$ . What is the coefficient of $x$ ?
Expand $(x + 1)(x + 5)$ . What is the coefficient of $x$ ?
Expand $(x + 7)(x + 3)$ . What is the coefficient of $x$ ?

Watch out!

Practice ★★

What is the constant term of $(x + 9)(x^2 + 1x + 6)$ ?
What is the constant term of $(x + 7)(x^2 + 7x + 3)$ ?

A polynomial of degree 5 is multiplied by a polynomial of degree 5. What is the degree of the product?

Exit ticket

Three questions to check you have got it.

Expand $(x + 6)(x + 9)$ . What is the coefficient of $x$ ?
Expand $(x + 1)(x + 5)$ . What is the coefficient of $x$ ?
Expand $(x + 7)(x + 3)$ . What is the coefficient of $x$ ?

Summary card

Polynomial arithmetic

Add: combine like terms.
Multiply: every term by every term; degrees add.

Watch out: $x$ -terms add; constants multiply.

2. 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.

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. Dividing polynomials

Warm-up

Three quick questions from earlier lessons.

Discover

$x^2 + 5x + 6 = (x + 2)(\ ?\ )$ . What goes in the gap?

$x + 3$
$x + 2$

Key idea

Long division works for polynomials too.

Divide the leading term, multiply back, subtract, bring down, repeat. Write missing powers with coefficient 0. Result:

p(x) = d(x)\,q(x) + r(x)

, with the remainder

r

of lower degree than

d

. Dividing by

x - a

, the remainder is

p(a)

Worked example

$(x^3 - 8) \div (x - 2)$ .

2x^2 + 0x - 8

x³ ÷ x = x²; subtract x²(x − 2) = x³ − 2x².

4x - 8

2x² ÷ x = 2x; subtract 2x(x − 2).

x^3 - 8 = (x - 2)(x^2 + 2x + 4)

4x ÷ x = 4; remainder 0.

Your turn, guided

Your turn: the remainder of $(x^2 + x + 1) \div (x - 1)$ .

3

Evaluate at 1.

Practice ★

Divide: $\dfrac{x^2 + 13x + 42}{x + 6}$ . Enter the quotient.
Divide: $\dfrac{x^2 + 12x + 35}{x + 7}$ . Enter the quotient.
Divide: $\dfrac{x^2 + 10x + 16}{x + 2}$ . Enter the quotient.

Watch out!

Practice ★★

Divide $x^3 - 1$ by $x - 1$ . Enter the quotient.
Divide $x^3 - 27$ by $x - 3$ . Enter the quotient.

Find the remainder when $x^2 + 4x + 1$ is divided by $x - 4$ .

Boss problem. $p(x) = x^3 - 8x^2 + 17x - 10$ has the factor $x - 1$ .

Exit ticket

Three questions to check you have got it.

Divide: $\dfrac{x^2 + 13x + 42}{x + 6}$ . Enter the quotient.
Divide: $\dfrac{x^2 + 12x + 35}{x + 7}$ . Enter the quotient.
Divide: $\dfrac{x^2 + 10x + 16}{x + 2}$ . Enter the quotient.

Summary card

Polynomial division

$p = d \cdot q + r$ ; keep 0-coefficients for missing powers.

Watch out: check by multiplying back.

5. 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 $0$ .

6. Polynomial identities

Warm-up

Three quick questions from earlier lessons.

Discover

Without a calculator: $51^2 - 49^2$ . Which shortcut helps?

$(51 + 49)(51 - 49) = 200$
$(51 - 49)^2 = 4$

Key idea

Identities are true for every value of the variables.

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

;

a^3 + b^3 = (a + b)(a^2 - ab + b^2)

;

a^3 - b^3 = (a - b)(a^2 + ab + b^2)

;

(1 - r)(1 + r + \dots + r^{n-1}) = 1 - r^n

Worked example

Factor $x^3 + 27$ .

a = x

b = 3

27 = 3³.

(x + 3)(x^2 - 3x + 9)

Sum of cubes.

The middle terms cancel

Check one term: x · 9 + 3 · (−3x) = 0x.

Your turn, guided

Your turn: $1 + 3 + 9 + 27$ with $r = 3$ , $n = 4$ : $(1 - 3^4)/(1 - 3)$ .

40

(1 − 81)/(−2).

Practice ★

Use an identity to compute $33^2 - 27^2$ .
Use an identity to compute $96^2 - 86^2$ .

$x^3 + 64 = (x + 4)(x^2 + bx + 16)$ . Find $b$ .

Watch out!

Practice ★★

Use $(1 - r)(1 + r + r^2 + \dots + r^{n-1}) = 1 - r^n$ to compute $1 + 2 + 4 + \dots + 2^{ 4 }$ .
Use $(1 - r)(1 + r + r^2 + \dots + r^{n-1}) = 1 - r^n$ to compute $1 + 2 + 4 + \dots + 2^{ 9 }$ .

Exit ticket

Three questions to check you have got it.

Use an identity to compute $33^2 - 27^2$ .
Use an identity to compute $96^2 - 86^2$ .
Use an identity to compute $23^2 - 17^2$ .

Summary card

Polynomial identities

$a^2 - b^2$ , $a^3 \pm b^3$ , and the geometric-sum identity.

Watch out: the sign pattern in the cube identities.

7. 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.

8. 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.

9. The binomial theorem

Warm-up

Three quick questions from earlier lessons.

Discover

$(1 + x)^2 = 1 + 2x + x^2$ and $(1 + x)^3 = 1 + 3x + 3x^2 + x^3$ . The coefficients $1, 3, 3, 1$ are a row of Pascal's triangle. What is the next row?

$1, 4, 6, 4, 1$
$1, 4, 4, 4, 1$

Key idea

The binomial theorem expands any power of a sum.

(a + b)^n = \displaystyle\sum_{k=0}^{n} \binom{n}{k} a^{\,n-k} b^{\,k}

, with

\dbinom{n}{k} = \dfrac{n!}{k!\,(n - k)!}

. For example

\dbinom{n}{1} = n

and

\dbinom{n}{2} = \dfrac{n(n - 1)}{2}

Worked example

Find the coefficient of $x^2$ in $(x + 3)^5$ .

Term:

\dbinom{5}{3} x^2 \cdot 3^3

x² means b = 3 appears 3 times: k = 3 with b = 3.

10 \cdot 27 = 270

C(5, 3) = 10.

Your turn, guided

Your turn: the coefficient of $x^2$ in $(1 + x)^6$ .

\dbinom{6}{2} = \dfrac{6 \cdot 5}{2}

C(6, 2).

= 15

30 ÷ 2.

Practice ★

Find the coefficient of $x^2$ in $(1 + x)^{ 10 }$ .
Find the coefficient of $x^2$ in $(1 + x)^{ 7 }$ .
Find the coefficient of $x^2$ in $(1 + x)^{ 5 }$ .

Watch out!

Practice ★★

Find the coefficient of $x$ in $(x + 4)^{ 4 }$ .
Find the coefficient of $x$ in $(x + 3)^{ 5 }$ .
Find the coefficient of $x$ in $(x + 5)^{ 4 }$ .

In $(2x + 2)^2 = 4x^2 + kx + 4$ , what is $k$ ?
In $(7x + 2)^2 = 49x^2 + kx + 4$ , what is $k$ ?

Exit ticket

Three questions to check you have got it.

Find the coefficient of $x$ in $(x + 4)^{ 4 }$ .
Find the coefficient of $x$ in $(x + 3)^{ 5 }$ .
Find the coefficient of $x$ in $(x + 5)^{ 4 }$ .

Summary card

The binomial theorem

$(a + b)^n = \sum \dbinom{n}{k} a^{\,n-k} b^{\,k}$ .
Coefficients: a row of Pascal's triangle.
Include the powers of the constants.

Watch out: $(a + b)^2 \ne a^2 + b^2$ .

10. 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.

11. 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.

12. Solving rational equations

Warm-up

Three quick questions from earlier lessons.

Discover

Solve $\dfrac{6}{x} = 2$ . Which value works?

$x = 3$
$x = 12$

Key idea

Clear the denominators, solve, then check.

Multiply both sides by the common denominator, solve the resulting polynomial equation, then check every solution in the original equation: a value that makes a denominator 0 is extraneous and must be rejected.

Worked example

$\dfrac{x}{x - 2} = \dfrac{2}{x - 2} + 1$ .

x = 2 + (x - 2)

, so

x = x

Multiply by x − 2.

…except where the denominator is 0

True for every x.

Solutions: every

x \ne 2

x = 2 is excluded.

Your turn, guided

Your turn: $\dfrac{x}{x + 1} = \dfrac{2}{3}$ .

x = 2

3x = 2x + 2.

Practice ★

Solve $\dfrac{x}{x + 2} = \dfrac{ 2 }{ 7 }$ .
Solve $\dfrac{x}{x + 2} = \dfrac{ 4 }{ 6 }$ .
Solve $\dfrac{x}{x + 6} = \dfrac{ 5 }{ 8 }$ .

Watch out!

Practice ★★

Solve $\dfrac{1}{x} + \dfrac{1}{ 11 } = \dfrac{1}{ 5 }$ .
Solve $\dfrac{1}{x} + \dfrac{1}{ 12 } = \dfrac{1}{ 2 }$ .

Solve $\dfrac{x^2 - 4}{x - 2} = 4$ .
1. No solution: simplifying gives x = 2, but x = 2 makes the denominator 0
2. x = 2
3. Every x except 2

Exit ticket

Three questions to check you have got it.

Solve $\dfrac{x}{x + 2} = \dfrac{ 2 }{ 7 }$ .
Solve $\dfrac{x}{x + 2} = \dfrac{ 4 }{ 6 }$ .
Solve $\dfrac{x}{x + 6} = \dfrac{ 5 }{ 8 }$ .

Summary card

Rational equations

Clear denominators → solve → check in the original.

Watch out: reject values that make a denominator 0.

Answers: Polynomials and rational functions

1. Adding, subtracting and multiplying polynomials

$15$
$6$
$10$

$54$
$21$

$10$

$15$
$6$
$10$

2. 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 $x \to \pm\infty$ , $f(x) \to -\infty$

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

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

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. Dividing polynomials

$x + 7$
$x + 5$
$x + 8$

$x^{2} + x + 1$
$x^{2} + 3 x + 9$

$33$

$x + 7$
$x + 5$
$x + 8$

5. The remainder theorem and the factor theorem

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

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

$-4$
$-3$

$19$
$27$
$-3$

6. Polynomial identities

$360$
$1820$

$-4$

$31$
$1023$

$360$
$1820$
$240$

7. Rational functions: vertical and horizontal asymptotes

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

$y = 0$
$y = 0$

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

8. Rational functions: intercepts and holes

$4$
$14$

$4$
$14$
$10$

9. The binomial theorem

$45$
$21$
$10$

$256$
$405$
$500$

$8$
$28$

$256$
$405$
$500$

10. 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

11. Adding and subtracting rational expressions

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

$x^{2} - 16$

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

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

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

12. Solving rational equations

$\frac{4}{5}$
$4$
$10$

$\frac{55}{6}$
$\frac{12}{5}$

No solution: simplifying gives x = 2, but x = 2 makes the denominator 0

$\frac{4}{5}$
$4$
$10$