Zeros of polynomials & their graphs (article)

What you will learn in this lesson

When studying polynomials, you often hear the terms zeros, roots, factors and

x

-intercepts.

In this article, we will explore these characteristics of polynomials and the special relationship that they have with each other.

Fundamental connections for polynomial functions

For a polynomial

f

and a real number

k

, the following statements are equivalent:

$x = k$ ‍ is a root, or solution, of the equation $f (x) = 0$ ‍
$k$ ‍ is a zero of function $f$ ‍
$(k, 0)$ ‍ is an $x$ ‍-intercept of the graph of $y = f (x)$ ‍
$x - k$ ‍ is a linear factor of $f (x)$ ‍

Let's understand this with the polynomial

g (x) = (x - 3) (x + 2)

, which can be written as

g (x) = (x - 3) (x - (- 2))

First, we see that the linear factors of

g (x)

are

(x - 3)

and

(x - (- 2))

If we set

g (x) = 0

and solve for

x

, we get

x = 3

x = - 2

. These are the solutions, or roots, of the equation.

A zero of a function is an

x

-value that makes the function value

0

. Since we know

x = 3

and

x = - 2

are solutions to

g (x) = 0

, then

3

and

- 2

are zeros of the function

g

Finally, the

x

-intercepts of the graph of

y = g (x)

satisfy the equation

0 = g (x)

, which was solved above. The

x

-intercepts of the equation are

(3, 0)

and

(- 2, 0)

Check your understanding

1) What are the zeros of $f (x) = (x + 4) (x - 7)$ ‍?

2) The graph of function $g$ ‍ crosses the $x$ ‍-axis at $(2, 0)$ ‍. What must be a root of the equation $g (x) = 0$ ‍?

x =

3) The zeros of function $h$ ‍ are $- 1$ ‍ and $3$ ‍. Which of the following could be $h (x)$ ‍?

Zeros and multiplicity

When a linear factor occurs multiple times in the factorization of a polynomial, that gives the related zero multiplicity.

For example, in the polynomial

f (x) = (x - 1) (x - 4)^{2}

, the number

4

is a zero of multiplicity

2

Notice that when we expand

f (x)

, the factor

(x - 4)

is written

2

times.

f (x) = (x - 1) (x - 4) (x - 4)

So in a sense, when you solve

f (x) = 0

, you will get

x = 4

twice.

\begin{aligned} 0 & = (x - 1) (x - 4) (x - 4) \\ x - 1 = 0 x - 4 = 0 x - 4 = 0 \\ x = 1 x = 4 x = 4 \end{aligned}

In general, if

x - k

occurs

m

times in the factorization of a polynomial, then

k

is a zero of multiplicity

m

. A zero of multiplicity

2

is called a double zero.

Check your understanding

4) Which zero of $f (x) = (x - 3) (x - 1)^{3}$ ‍ has multiplicity $3$ ‍?

5) Which zero of $g (x) = (x + 1)^{3} (2 x + 1)^{2}$ ‍ is a double zero?

The graphical connection

The multiplicity of a zero is important because it tells us how the graph of the polynomial will behave around the zero.

For example, notice that the graph of

f (x) = (x - 1) (x - 4)^{2}

behaves differently around the zero

1

than around the zero

4

, which is a double zero.

Specifically, while the graphs crosses the

x

-axis at

x = 1

, it only touches the

x

-axis at

x = 4

Let's look at the graph of a function that has the same zeros, but different multiplicities. For example, consider

g (x) = (x - 1)^{2} (x - 4)

. Notice that for this function

1

is now a double zero, while

4

is a single zero.

Now we see that the graph of

g

touches the

x

-axis at

x = 1

and crosses the

x

-axis at

x = 4

In general, if a function

f

has a zero of odd multiplicity, the graph of

y = f (x)

will cross the

x

-axis at that

x

value. If a function

f

has a zero of even multiplicity, the graph of

y = f (x)

will touch the

x

-axis at that point.

Check your understanding

6) In the graphed function, is the multiplicity of the zero $6$ ‍ even or odd?

7) Which is the graph of $h (x) = x^{2} (x - 3)$ ‍?

Challenge problem

8*) Which is the graph of $f (x) = - x^{3} + 4 x^{2} - 4 x$ ‍?

Course: Algebra (all content) > Unit 10

Zeros of polynomials & their graphs

What you will learn in this lesson

Fundamental connections for polynomial functions

Check your understanding

Zeros and multiplicity

Check your understanding

The graphical connection

Check your understanding

Challenge problem

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