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4. Polynomial Functions & Expressions
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Algebra 2

4.04 Zeros of polynomial functions

Lesson
Practice
Lesson

Concept summary

The zeros of a function are the input values which make the function equal to zero. This means that zeros are the solutions to the equation f\left(x\right) = 0.

The multiplicity of a zero is the number of times that its corresponding factor appears in the function. Zeros with different multiplicities look different graphically.

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y
Zeros of multiplicity 1, cuts through axis with no point of inflection
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y
Zeros of multiplicity 2, bounces off the axis
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y
Zeros of multiplicity 3, cuts through the axis with a point of inflection
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y
Graph of y=(x+3)(x+1)^2(x-2)^3 with zeros of multiplicities 1, 2, and 3
End behavior

Describes the trend of a function or graph at its left and right ends; specifically the y-value that each end obtains or approaches.

x-intercept

A point where a line or graph intersects the x-axis. The value of y is 0 at this point.

The line y=-x+3 drawn in a coordinate plane. The point (3,0) is labeled x-intercept
y-intercept

A point where a line or graph intersects the y-axis. The value of x is 0 at this point.

The line y=-x+3 graphed in the coordinate plane. The point (0,3) is labeled y-intercept

Worked examples

Example 1

Consider the function f(x)=x^3-4x^2-8x+8, with a zero at x=3-\sqrt{5}.

a

Determine all the zeros of f(x) and their multiplicities.

Approach

To find the zeros of the function f(x), we set f(x)=0 and solve this equation for x. Each value of x is a zero of the function.

To determine the multiplicities of each zero, we determine the number of times that the corresponding factor of each zero occurs in the function.

We can use the fact that we have a radical root to identify another using its conjugate.

Solution

Since we are told that x=3-\sqrt{5} is a zero, this means that its conjugate x=3+\sqrt{5} must also be a zero.

This function has degree 3, so has at most one more zero.

Note that if a number k is a zero of f(x), then f(k)=0. It follows that x-k is a factor of f(x).

The factors of the constant term, 8, are \pm1,\pm2,\pm4,\pm8 and the factors of the leading coefficient 1 only includes \pm 1, so these are all the possible factors.

Observe that

  • f\left(-2\right)=\left(-2\right)^3-4\left(-2\right)^2-8\left(-2\right)+8=0

Therefore the zeros of x^3-4x^2-8x+8 are x=3 \pm \sqrt{5} and x=-2.

Since the function has degree 3 and 3 unique zeros, each zero has multiplicity of 1.

b

Determine the end behavior of f\left(x\right) as x\to\infty.

Approach

We determine first the leading coefficient of f(x)=x^3-4x^2-8x+8 and its degree either odd or even since the end behavior of a polynomial function is determined by its degree and leading coefficient.

Solution

The function f(x)=x^3-4x^2-8x+8 has a positive leading coefficient which is 1.

Note that if the polynomial function has a positive leading coefficient, the end behavior is as follows:

Even degreeOdd degree
\text{As }x\to -\infty, f(x)\to \infty.\text{As }x\to -\infty, f(x)\to -\infty.
\text{As }x\to \infty, f(x)\to \infty.\text{As }x\to \infty, f(x)\to \infty.

The degree of f(x) is 3 which is odd.

Therefore, as x\to \infty, f(x)\to \infty.

c

Determine the end behavior of f\left(x\right) as x\to -\infty.

Solution

The function f(x)=x^3-4x^2-8x+8 has a positive leading coefficient which is 2.

Note that if the polynomial function has a positive leading coefficient, the end behavior is as follows:

Even degreeOdd degree
\text{As }x\to -\infty, f(x)\to \infty.\text{As }x\to -\infty, f(x)\to -\infty.
\text{As }x\to \infty, f(x)\to \infty.\text{As }x\to \infty, f(x)\to \infty.

The degree of f(x) is 3 which is odd.

Therefore, as x\to -\infty, f(x)\to -\infty.

Reflection

Another way of determining the end behavior of f(x)=x^3-4x^2-8x+8=\left(x+2\right)\left(x-3+\sqrt{}5\right)\left(x-3-\sqrt{5}\right) is through looking at its graph.

-3
-2
-1
1
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3
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5
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7
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-30
-25
-20
-15
-10
-5
5
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35
f(x)

Example 2

Sketch a graph of the function f(x)=-3(x+3)(2x+1)^2, labeling the intercepts.

Approach

To label the y-intercept in the graph, we first find the value of f(x) at x=0.

To label the x-intercept in the graph, we first find the value of x such that f(x)=0. The x-values of the x-intercepts are the zeros of the function.

Solution

Let x=0. Substituting x=0 into the function, we have f(0)=-3(0+3)\left(2(0)+1\right)^2=-9 The y-intercept is (0,-9).

Let f(x)=0. Then we have -3(x+3)(2x+1)^2=0 This implies that x+3=0 and 2x+1=0 which gives x=-3 and x=-\dfrac{1}{2}, respectively. The coordinates of the x-intercepts are (-3,0) with multiplicity 1 and \left(-\dfrac{1}{2},0\right) with multiplicity 2. That means the graph will cut through the x-axis at x=-3 and bounce off the x-axis at x=-\frac{1}{2}.

The leading coefficient is negative so the end behavior will be up to the left and down to the right.

The graph of the function is shown below.

x
f(x)

Example 3

A polynomial function f(x) has the following characteristics:

  • Rational coefficients

  • Degree of 3

  • Zeros include x=3 with multiplicity 1 and x=-\sqrt{2} with multiplicity 1

  • Leading coefficient of 2

Sketch a graph of the function f(x), including all intercepts.

Approach

To graph the polynomial function f(x), we first find this function in factored form based on the given information.

From there we can find the y-intercept and identify the end behavior.

Solution

The zeros of the function that we know and their multiplicities are x=3 with multiplicity 1 and x=-\sqrt{2} with multiplicity 1. If x=-\sqrt{2} is a zero, then x=\sqrt{2} must also be a zero. From these, we get the expression (x-3)\left(x-\sqrt{2}\right)\left(x+\sqrt{2}\right)

Clearly, the degree of the product of the factors is 3. Since the leading coefficient of is 2, f(x)=2(x-3)\left(x-\sqrt{2}\right)\left(x+\sqrt{2}\right) is the polynomial function.

We can now find the y-intercept using the equation:

\displaystyle f(x)\displaystyle =\displaystyle 2(x-3)\left(x-\sqrt{2}\right)\left(x+\sqrt{2}\right)Equation of polynomial
\displaystyle f(0)\displaystyle =\displaystyle 2(0-3)\left(0-\sqrt{2}\right)\left(0+\sqrt{2}\right)Substitute x=0
\displaystyle f(0)\displaystyle =\displaystyle 12Evaluate the parentheses and products

The y-intercept is \left(0,12\right).

We will now determine the end behavior of f(x)=2(x-3)\left(x-\sqrt{2}\right)\left(x+\sqrt{2}\right).

The degree of the function is 3 which is odd and the leading coefficient is 2 which is positive.

So as x\to\infty, we have f(x)\to\infty and as x\to-\infty, we have f(x)\to-\infty.

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f(x)

Outcomes

A2.A.APR.A.2

Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial.

A2.MP1

Make sense of problems and persevere in solving them.

A2.MP2

Reason abstractly and quantitatively.

A2.MP3

Construct viable arguments and critique the reasoning of others.

A2.MP6

Attend to precision.

A2.MP7

Look for and make use of structure.

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