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

3. 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 $2a$ .

4. 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.

5. 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 $x = -3$ .

6. 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.

7. 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 $0$ .

8. 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 $-3$ .

9. 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.

10. 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.

Answers: Polynomial relationships

1. Adding, subtracting and multiplying polynomials

$15$
$6$
$10$

$54$
$21$

$10$

$15$
$6$
$10$

2. The binomial theorem

$45$
$21$
$10$

$256$
$405$
$500$

$8$
$28$

$256$
$405$
$500$

3. Quadratic equations with complex solutions

$6 i$
$9 i$

$33$

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

$5 + 2 i$

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

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

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

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

5. 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

$0$ · $9$ · $-7$
$0$ · $2$ · $-4$

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

6. Dividing polynomials

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

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

$33$

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

7. The remainder theorem and the factor theorem

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

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

$-4$
$-3$

$19$
$27$
$-3$

8. 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

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

9. Polynomial identities

$360$
$1820$

$-4$

$31$
$1023$

$360$
$1820$
$240$

10. Complex zeros and conjugate pairs

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

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

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

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