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Zeros and Successive Differences of Polynomial Functions (College Board AP® Precalculus): Study Guide

Written by: Roger B

Reviewed by: Mark Curtis

Updated on 9 April 2026

Zeros of polynomial functions

What is a zero of a polynomial function?

If $a$ is a number and $p (a) = 0$
- then $a$ is called a zero of the polynomial function $p$
- Equivalently, $a$ is a root of the equation $p (x) = 0$
Zeros of polynomial functions can be real or complex (non-real) numbers

If $a$ is a real number, then $(x - a)$ is a linear factor of $p$ if and only if $a$ is a zero of $p$
- I.e. if $(x - a)$ is a linear factor of $p$ , then $p (a) = 0$
- Or if $p (a) = 0$ , then $(x - a)$ is a linear factor of $p$
E.g. if $p (x) = (x - 3) (x + 1) (x - 5)$ , then the zeros of $p$ are $x = 3$ , $x = - 1$ , and $x = 5$
- This lets you write down the solutions to $p (x) = 0$ if you have $p (x)$ in fully factorised form
Conversely, if you know that $x = 2$ is a zero of a polynomial, then $(x - 2)$ must be a factor

What is the multiplicity of a zero?

If a linear factor $(x - a)$ is repeated $n$ times in the factorisation of a polynomial
- the corresponding zero has multiplicity $n$
E.g. $p (x) = (x - 1)^{2} (x + 4)^{3} (x + 7)$ has
- a zero at $x = 1$ with multiplicity 2
- a zero at $x = - 4$ with multiplicity 3
- and a zero at $x = - 7$ with multiplicity 1
A polynomial function of degree $n$ has exactly $n$ complex zeros when counting multiplicities
- A 'complex zero' here means a zero of the function that is either a real or a (non-real) complex number
E.g. a degree 5 polynomial always has exactly 5 zeros (counting multiplicities)
- though some may be repeated (i.e. have a multiplicity greater than 1)
- and some may be complex (i.e. non-real)

How do zeros appear on a graph?

If $a$ is a real zero of a polynomial function $p$ , then the graph of $y = p (x)$ has an $x$ -intercept at the point $(a, 0)$
- The graph crosses or touches the $x$ -axis at that point
The multiplicity of a zero determines the behaviour of the graph at that $x$ -intercept:
If the zero has odd multiplicity (e.g. 1, 3, 5, ...), the graph crosses the $x$ -axis at that point
- The sign of the output values changes on either side of the zero
- If the multiplicity is 1, the graph 'cuts straight across' the $x$ -axis
- If the multiplicity is 3, 5 or more, then the curve 'levels out' as it crosses the $x$ -axis
If the zero has even multiplicity (e.g. 2, 4, 6, ...), the graph is tangent to (touches but does not cross) the $x$ -axis at that point
- The sign of the output values is the same on both sides of the zero
- The graph touches the axis then 'turns back'
Graph behavior at zeros with different multiplicities can be seen here:

Graph of y=(x+2)²(x−1)(x-4)³ with x-intercepts at -2, 1, and 4, indicating multiplicities 2, 1, and 3 respectively, on xy-plane. — **Zeros with different multiplicities on the graph of a polynomial function**

How can zeros help solve polynomial inequalities?

The real zeros of a polynomial are the points where the output value is zero
Therefore they divide the $x$ -axis into intervals
- On each interval between consecutive zeros, the polynomial is either entirely positive or entirely negative
- This means the real zeros act as endpoints for intervals satisfying polynomial inequalities
E.g. to solve $p (x) \geq 0$
- find the zeros first
- then determine the sign of $p (x)$ on each interval between zeros

How can I find the zeros of a polynomial?

If the polynomial is given in factored form, set each factor equal to zero and solve
- E.g. for $p (x) = (x - 3) (x + 1) (x - 5) = 0$
  - $x - 3 = 0 \Rightarrow x = 3$
  - $x + 1 = 0 \Rightarrow x = - 1$
  - $x - 5 = 0 \Rightarrow x = 5$
- The zeros are $x = 3$ , $x = - 1$ , and $x = 5$
If the polynomial is a quadratic, you can factor it (if possible) or use the quadratic formula
For higher-degree polynomials that are not easily factored, you can use a graphing calculator to find the real zeros
- Enter the polynomial into the calculator and find where the graph crosses the $x$ -axis
- Or use the calculator's "zero" or "solve" function
- E.g. to find the zeros of $g (x) = - 0.167 x^{3} + x^{2} - 1.834$
  - you would enter the function into your calculator
  - and find that the zeros are approximately $x = - 1.233$ , $x = 1.578$ , and $x = 5.643$

Examiner Tips and Tricks

The zeros found on your calculator may be decimal approximations.

On your exam, report these accurate to three decimal places

What about complex (non-real) zeros?

Some zeros of a polynomial may be complex numbers
- I.e. numbers involving $i = \sqrt{- 1}$
These do not correspond to $x$ -intercepts on the graph
- since only real zeros appear on the real number line (i.e. on the $x$ -axis)
Complex zeros always come in conjugate pairs
- I.e. if $a + b i$ is a zero of a polynomial $p$ (where $b \neq 0$ )
  - then $a - b i$ is also a zero of $p$
- E.g. if $2 + 3 i$ is a zero, then $2 - 3 i$ must also be a zero
This means non-real zeros always account for an even number (0, 2, 4, 6...) of the total zeros

Examiner Tips and Tricks

When a question gives you a polynomial with a non-real complex zero, immediately write down its conjugate.

That gives you a second zero for free!

This is particularly useful for determining how many real zeros (and therefore $x$ -intercepts) a polynomial has

For example, a degree 4 polynomial with a given non-real zero must also have the conjugate as a zero
That leaves only 2 remaining zeros, which may be real or another non-real conjugate pair

Worked Example

The polynomial function $p$ is given by $p (x) = - x (x - 4)^{2} (x + 2)$ .

(a) Find all zeros of $p$ and state the multiplicity of each zero.

Answer:

Set each factor equal to zero:

$- x = 0 \Rightarrow x = 0$

$(x - 4)^{2} = 0 \Rightarrow x - 4 = 0 \Rightarrow x = 4$

$x + 2 = 0 \Rightarrow x = - 2$

The 'squared' (power of 2) on ${(x - 4)}^{2}$ tells you that $x = 4$ has a multiplicity of 2

The other two zeros are both multiplicity 1

$x = 0 (multiplicity 1)$

$x = 4 (multiplicity 2)$

$x = - 2 (multiplicity 1)$

(b) For each zero, state whether the graph of $y = p (x)$ crosses the $x$ -axis or is tangent to the $x$ -axis at that point.

Answer:

Use the multiplicity to determine the behavior of the graph at each zero

At $x = 0$ :
multiplicity is $1$ (odd), so the graph crosses the $x$ -axis

At $x = 4$ :
multiplicity is $2$ (even), so the graph is tangent to the $x$ -axis
(touches but does not cross)

At $x = - 2$ :
multiplicity is $1$ (odd), so the graph crosses the $x$ -axis

Answer:

The zeros divide the $x$ -axis into four intervals: $(- \infty, - 2)$ , $(- 2, 0)$ , $(0, 4)$ , and $(4, \infty)$

Test a value in each interval to determine the sign of $p (x)$

$x = - 3$ :
$\begin{array}{rcl} p (- 3) & = & - (- 3) (- 3 - 4)^{2} (- 3 + 2) \\ = & - (- 3) (49) (- 1) \\ = & - 147 < 0 \end{array}$

$x = - 1$ :
$\begin{array}{rcl} p (- 1) & = & - (- 1) (- 1 - 4)^{2} (- 1 + 2) \\ = & (1) (25) (1) \\ = & 25 > 0 \end{array}$

$x = 2$ :
$\begin{array}{rcl} p (2) & = & - (2) (2 - 4)^{2} (2 + 2) \\ = & - (2) (4) (4) \\ = & - 32 < 0 \end{array}$

$x = 5$ :
$\begin{array}{rcl} p (5) & = & - (5) (5 - 4)^{2} (5 + 2) \\ = & - (5) (1) (7) \\ = & - 35 < 0 \end{array}$

The polynomial is only positive on one of those intervals

$p (x) > 0$ on $(- 2, 0)$

Worked Example

The polynomial function $k$ is given by $k (x) = a x^{4} - b x^{3} + 22$ , where $a$ and $b$ are nonzero real constants. Each of the zeros of $k$ has multiplicity 1. In the $x y$ -plane, an $x$ -intercept of the graph of $k$ is $(12.405, 0)$ . A zero of $k$ is $1.362 - 0.547 i$ . Which of the following statements must be true?

(A) The graph of $k$ has three $x$ -intercepts.

(B) $1.362 + 0.547 i$ is a zero of $k$ .

(D) The graph of $k$ is tangent to the $x$ -axis at $x = 12.405$ .

Answer:

Because complex zeros come in conjugate pairs, if $1.362 - 0.547 i$ is a zero of $k$ , then $1.362 + 0.547 i$ must also be a zero of $k$

That already tells you that (B) is the correct answer
but it's worth considering why the other answers are not correct

$k$ has degree 4, so it has exactly 4 zeros (counting multiplicities)

Each zero corresponds to a solution to $k (x) = 0$
$1.362 - 0.547 i$ and $1.362 + 0.547 i$ are both non-real zeros
So the equation $k (x) = 0$ cannot have four real solutions
- That rules out option (C)

Two of the zeros are non-real

So the remaining 2 zeros must either both be real
or both be non-real
- because non-real zeros always occur in conjugate pairs
So $k$ can have at most 2 real zeros
which means at most 2 $x$ -intercepts
- So option (A) cannot be true

Finally, you are told that each zero has multiplicity 1

1 is odd so the graph crosses the $x$ -axis at each real zero, rather than being tangent
- This rules out option (D)

(B) $1.362 + 0.547 i$ is a zero of $k$

Degree and successive differences

How can successive differences reveal the degree of a polynomial?

The degree of a polynomial function can be found by examining the successive differences of the output values over equally spaced input values
To do this:
- Start with output values at equally spaced inputs
- Calculate the 1st differences (differences between consecutive outputs)
  - If the 1st differences are constant, the polynomial has degree 1 (linear)
- If not, calculate the 2nd differences (differences between consecutive 1st differences)
  - If the 2nd differences are constant, the polynomial has degree 2 (quadratic)
- If not, continue taking successive differences until you reach a constant set of differences
  - The degree of the polynomial is equal to the least value $n$ for which the $n$ th differences are constant
E.g. consider these values at equally spaced inputs for a polynomial function $p$ :

$x$	0	1	2	3	4
$p (x)$	2	5	14	35	74

Calculate the differences
- 1st differences: $3, 9, 21, 39$ — not constant
- 2nd differences: $6, 12, 18$ — not constant
- 3rd differences: $6, 6$ — constant

Pattern of numbers showing sequence 2, 5, 14, 35, 74. First, second, and third differences are illustrated.

The 3rd differences are constant, so $p$ is a degree 3 (cubic) polynomial

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Zeros and Successive Differences of Polynomial Functions (College Board AP® Precalculus): Study Guide

Zeros of polynomial functions

What is a zero of a polynomial function?

What is the multiplicity of a zero?

How do zeros appear on a graph?

How can zeros help solve polynomial inequalities?

How can I find the zeros of a polynomial?

Examiner Tips and Tricks

What about complex (non-real) zeros?

Examiner Tips and Tricks

Worked Example

Worked Example

Degree and successive differences

How can successive differences reveal the degree of a polynomial?

Unlock more, it's free!

Join the 100,000+ Students that ❤️ Save My Exams

Author: Roger B

Reviewer: Mark Curtis

Zeros and Successive Differences of Polynomial Functions (College Board AP® Precalculus): Study Guide

Zeros of polynomial functions

What is a zero of a polynomial function?

How are zeros related to factors?

What is the multiplicity of a zero?

How do zeros appear on a graph?

How can zeros help solve polynomial inequalities?

How can I find the zeros of a polynomial?

What about complex (non-real) zeros?

Degree and successive differences

How can successive differences reveal the degree of a polynomial?

Unlock more, it's free!

Join the 100,000+ Students that ❤️ Save My Exams

Author: Roger B

Reviewer: Mark Curtis