Polynomials and Complex Zeros — AP® Precalculus

Part 1 · Hook

Why this matters

An engineer reading a vibration signal wants to know where it is zero and whether it passes through zero or just grazes it. For polynomial functions, the factors answer both questions: each factor gives a zero, and its power says whether the graph crosses or touches. Some zeros are not even real, but they still count.

Part 2 · Before you start

What this builds on

Part 3 · Prerequisite check

Quick check before you start

1. Solve $x^2 + 4 = 0$ .

$x = \pm 2i$
$x = \pm 2$
$x = \pm 4i$
$x = -4$

Show the answer

$x^2 = -4$ , so $x = \pm\sqrt{-4} = \pm 2i$ .

Correct: $x = \pm 2i$ :
$x = \pm 2$ :
$x = \pm 4i$ :
$x = -4$ :

2. What is the degree of $-2(x - 1)(x + 3)^2$ ?

$3$
$2$
$-2$
$4$

Show the answer

$x \cdot x^2 = x^3$ .

Correct: $3$ :
$2$ :
$-2$ :
$4$ :

Part 4 · See it

See it first

The graph of p(x) = 0.25(x + 2)(x − 1)²(x − 3). It crosses the x-axis at x = −2 and x = 3, the zeros of multiplicity 1, and touches the axis without crossing at x = 1, the zero of multiplicity 2. — $p(x)$ $= 0.25(x + 2)(x - 1)^2(x - 3)$ crosses the $x$ -axis at the zeros of odd multiplicity, $-2$ and $3$ , and touches it at the zero of even multiplicity, $1$ . LevlPrep original diagram.

Part 5 · Step by step

How it works, step by step

$(x - a)^k$ is a factor of $p$ and no higher power is $a$ is a zero of multiplicity $k$
The multiplicity is even $p$ keeps its sign near $a$ : the graph touches the axis without crossing
$p$ has degree $n$ it has exactly $n$ complex zeros counting multiplicity, real or not
$p$ has real coefficients and a zero $a + bi$ with $b \ne 0$ its conjugate $a - bi$ is a zero too

Part 6 · Key ideas

Key ideas

Multiplicity: the power of the factor $(x - a)$ . Odd: the graph crosses. Even: it touches (a double zero bounces off the axis).
Degree $n$ means $n$ complex zeros counting multiplicity; non-real ones come in conjugate pairs when the coefficients are real.
Equally spaced inputs with constant $n$ th differences fit degree $n$ .
Even function: $f(-x) = f(x)$ , symmetric about the $y$ -axis. Odd function: $f(-x) = -f(x)$ , symmetric about the origin.

Part 7 · Misconception

A common mistake

The wrong idea: A polynomial of degree $4$ always has $4$ $x$ -intercepts.

What actually happens: It has $4$ complex zeros counting multiplicity, but some may repeat and some may be non-real. The $p$ in the figure has degree $4$ and only $3$ $x$ -intercepts, because the zero $1$ counts twice.

Part 8 · Check yourself

Check yourself

Exam-style questions. Anything you miss goes into your review queue.

Graph

The graph of a polynomial function $p$

1. The graph shows a polynomial function $p$ of degree $4$ . Which zero of $p$ has even multiplicity?

$x = -3$
$x = 2$
$x = -1$
$x = 0$

Show the answer

At a zero of even multiplicity the factor is squared (or raised to another even power), so it does not change sign: the graph touches the $x$ -axis without crossing.

$x = -3$ : The graph crosses the axis at $x = -3$ , which goes with odd multiplicity.
$x = 2$ : The graph crosses at $x = 2$ as well: odd multiplicity.
Correct: $x = -1$ : Right: the graph touches the axis at $x = -1$ and turns back without crossing.
$x = 0$ : $p(0) = 1.5$ , so $0$ is not a zero at all.

2. Which could be a formula for $p$ ?

$0.25(x + 3)(x + 1)^2(x - 2)$
$-0.25(x + 3)(x + 1)(x - 2)$
$-0.25(x - 3)(x - 1)^2(x + 2)$
$-0.25(x + 3)(x + 1)^2(x - 2)$

Show the answer

A zero at $x = a$ gives a factor $x - a$ . The touching zero $-1$ needs $(x + 1)^2$ . Then check one point: at $x = 0$ the formula must give the graph's $1.5$ .

$0.25(x + 3)(x + 1)^2(x - 2)$ : This has the right zeros but the opposite sign: its output at $0$ is $-1.5$ , below the axis.
$-0.25(x + 3)(x + 1)(x - 2)$ : Degree $3$ , and $x = -1$ appears once, so the graph would cross there.
$-0.25(x - 3)(x - 1)^2(x + 2)$ : These zeros are $3$ , $1$ and $-2$ , the opposites of the ones on the graph.
Correct: $-0.25(x + 3)(x + 1)^2(x - 2)$ : Right: zeros $-3$ , $-1$ (squared) and $2$ , and $p(0)$ $= -0.25(3)(1)(-2)$ $= 1.5$ matches the graph.

3. $p$ has degree $4$ . How many complex zeros does $p$ have, counting multiplicity?

No calculator. Type the exact value (a fraction, π, e or a root is fine).

Show the answer

A polynomial of degree $n$ has exactly $n$ complex zeros, counting multiplicity. Here: $-3$ (once), $-1$ (twice) and $2$ (once), $4$ in all, all of them real.

Answer: 4

Table

Equally spaced outputs

Values of f(x) = x^3 - 2x^2 + 1 at x = 0, 1, 2, 3, 4, 5
$x$	0	1	2	3	4	5
$f(x)$	1	0	1	10	33	76

4. For the outputs in the table, which differences are constant?

The first differences, each equal to $1$
The second differences, each equal to $6$
The third differences, each equal to $6$
No level of differences is constant

Show the answer

First differences $-1, 1, 9, 23, 43$ ; second $2, 8, 14, 20$ ; third $6, 6, 6$ . Constant third differences over equal steps fit a polynomial of degree $3$ .

The first differences, each equal to $1$ : The first differences are $-1, 1, 9, 23, 43$ .
The second differences, each equal to $6$ : The second differences are $2, 8, 14, 20$ ; they change by $6$ .
Correct: The third differences, each equal to $6$ : Right: $8 - 2 = 6$ , $14 - 8 = 6$ , $20 - 14 = 6$ .
No level of differences is constant: Keep differencing: the third differences settle at $6$ .

5. Using only the differences, what is $f(6)$ ?

No calculator. Type the exact value (a fraction, π, e or a root is fine).

Show the answer

Next third difference $6$ ; next second difference $20 + 6 = 26$ ; next first difference $43 + 26 = 69$ ; so $f(6) = 76 + 69 = 145$ . Check: $216 - 72 + 1 = 145$ .

Answer: 145

6. A polynomial function with real coefficients has $2 + i$ as a zero. Which must also be a zero?

$-2 - i$
$2 - i$
$-2 + i$
$i$

Show the answer

With real coefficients, the non-real zeros come in conjugate pairs $a \pm bi$ . So $2 - i$ is also a zero.

$-2 - i$ : This negates both parts; the conjugate changes only the sign of the imaginary part.
Correct: $2 - i$ : Right: non-real zeros of a polynomial with real coefficients come in conjugate pairs.
$-2 + i$ : The real part stays $2$ .
$i$ : Only the complex conjugate of $2 + i$ is forced to be a zero.

Part 9 · Summary

Summary

Each factor

(x - a)^k

of a polynomial gives a zero

a

of multiplicity

k

: the graph crosses at odd multiplicity and touches at even multiplicity. A polynomial of degree

n

has

n

complex zeros counting multiplicity, and with real coefficients the non-real ones come in conjugate pairs. Constant

n

th differences point to degree

n

; even and odd functions show mirror symmetry.

Part 10 · Up next

What comes next

Part 11 · Connections

Why this matters

What this builds on

Quick check before you start

See it first

How it works, step by step

Key ideas

A common mistake

Check yourself

The graph of a polynomial function pp

Equally spaced outputs

Summary

What comes next

Connections

The graph of a polynomial function $p$