Polynomial Functions: Learn It 4

Lumen Learning

Polynomial Functions: Learn It 4

Zeros of Polynomial Functions

Using the Rational Zero Theorem to Find Rational Zeros

Another use for the remainder theorem is to test whether a rational number is a zero for a given polynomial. But first we need a pool of rational numbers to test. The rational zero theorem helps us narrow down the number of possible rational zeros using the ratio of the factors of the constant term and factors of the leading coefficient of the polynomial.

Consider a quadratic function with two zeros, [latex]x = \frac{2}{5}[/latex] and [latex]x = \frac{3}{4}[/latex]. By the factor theorem, these zeros have factors associated with them. Let us set each factor equal to [latex]0[/latex], and then construct the original quadratic function absent its stretching factor.

[latex]\begin{array}{l l} x - \frac{2}{5} = 0 \text{ or } x - \frac{3}{4} = 0 & \text{Set each factor equal to 0.} \\ 5x - 2 = 0 \text{ or } 4x - 3 = 0 & \text{Multiply both sides of the equation to eliminate fractions.} \\ f(x) = (5x - 2)(4x - 3) & \text{Create the quadratic function, multiplying the factors.} \\ f(x) = 20x^2 - 23x + 6 & \text{Expand the polynomial.} \\ f(x) = (5 \cdot 4)x^2 - 23x + (2 \cdot 3) & \\ \end{array}[/latex]

Notice that two of the factors of the constant term, [latex]6[/latex], are the two numerators from the original rational roots: [latex]2[/latex] and [latex]3[/latex]. Similarly, two of the factors from the leading coefficient, [latex]20[/latex], are the two denominators from the original rational roots: [latex]5[/latex] and [latex]4[/latex].

We can infer that the numerators of the rational roots will always be factors of the constant term and the denominators will be factors of the leading coefficient. This is the essence of the rational zero theorem; it is a means to give us a pool of possible rational zeros.

the rational zero theorem

The Rational Zero Theorem states that, if the polynomial [latex]f(x) = a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0[/latex] has integer coefficients, then every rational zero of [latex]f(x)[/latex] has the form [latex]\frac{p}{q}[/latex] where [latex]p[/latex] is a factor of the constant term [latex]a_0[/latex] and [latex]q[/latex] is a factor of the leading coefficient [latex]a_n[/latex].

When the leading coefficient is [latex]1[/latex], the possible rational zeros are the factors of the constant term.

How to: Given a polynomial function [latex]f(x)[/latex], use the Rational Zero Theorem to find rational zeros.

Determine all factors of the constant term and all factors of the leading coefficient.
Determine all possible values of [latex]\frac{p}{q}[/latex], where [latex]p[/latex] is a factor of the constant term and [latex]q[/latex] is a factor of the leading coefficient. Be sure to include both positive and negative candidates.
Determine which possible zeros are actual zeros by evaluating each case of [latex]f\left(\frac{p}{q}\right)[/latex].

List all possible rational zeros of [latex]f(x) = 2x^4 - 5x^3 + x^2 - 4[/latex].

Finding the Zeros of Polynomial Functions

The rational zero theorem helps us to narrow down the list of possible rational zeros for a polynomial function. Once we have done this, we can use synthetic division repeatedly to determine all of the zeros of a polynomial function.

How to: Given a polynomial function [latex]f[/latex], use synthetic division to find its zeros.

Use the rational zero theorem to list all possible rational zeros of the function.
Use synthetic division to evaluate a given possible zero by synthetically dividing the candidate into the polynomial. If the remainder is [latex]0[/latex], the candidate is a zero. If the remainder is not zero, discard the candidate.
Repeat step two using the quotient found with synthetic division. If possible, continue until the quotient is a quadratic.
Find the zeros of the quadratic function. Two possible methods for solving quadratics are factoring and using the quadratic formula.

Find the zeros of [latex]f(x)=4x^3−3x−1[/latex].

The Rational Zero Theorem tells us that if [latex]\frac{p}{q}[/latex] is a zero of [latex]f(x)[/latex], then [latex]p[/latex] is a factor of [latex]-1[/latex] and [latex]q[/latex] is a factor of [latex]4[/latex].

[latex]\begin{array}{lcl}\frac{p}{q} & = &\frac{\text{factor of constant term}}{\text{factor of leading coefficient}} \\ & = & \frac{\text{factor of -1}}{\text{factor of 4}}\end{array}[/latex]

The factors of [latex]-1[/latex] are [latex]\pm1[/latex] and the factors of [latex]4[/latex] are [latex]\pm1, \pm2, \pm4[/latex]. The possible values for [latex]\frac{p}{q}[/latex] are [latex]\pm1, \pm\frac{1}{2}, \pm\frac{1}{4}[/latex]. These are the possible rational zeros for the function. We will use synthetic division to evaluate each possible zero until we find one that gives a remainder of [latex]0[/latex]. Let’s begin with [latex]1[/latex].

[latex]\begin{array}{c|cccc} 1 & 4 & 0 & -3 & -1 \\ & & 4 & 4 & 1 \\ \hline & 4 & 4 & 1 & 0 \\ \end{array}[/latex]

Dividing by [latex](x - 1)[/latex] gives a remainder of [latex]0[/latex], so [latex]1[/latex] is a zero of the function. The polynomial can be written as

[latex](x - 1)(4x^2 + 4x + 1)[/latex]

The quadratic is a perfect square. [latex]f(x)[/latex] can be written as

p[latex](x - 1)(2x + 1)^2[/latex]

We already know that [latex]1[/latex] is a zero. The other zero will have a multiplicity of [latex]2[/latex] because the factor is squared. To find the other zero, we can set the factor equal to [latex]0[/latex].

[latex]\begin{align*} 2x + 1 & = 0 \\ x & = -\frac{1}{2} \end{align*}[/latex]

The zeros of the function are [latex]1[/latex] and [latex]-\frac{1}{2}[/latex] with multiplicity [latex]2[/latex].

Analysis

Look at the graph of the function [latex]f[/latex] below. Notice, at [latex]x = -0.5[/latex], the graph bounces off the [latex]x[/latex]-axis, indicating the even multiplicity [latex](2,4,6...)[/latex] for the zero [latex]-0.5[/latex]. At [latex]x = 1[/latex], the graph crosses the [latex]x[/latex]-axis, indicating the odd multiplicity [latex](1,3,5...)[/latex] for the zero [latex]x = 1[/latex].

Graph of a polynomial that have its local maximum at (-0.5, 0) labeled as “Bounce” and its x-intercept at (1, 0) labeled, “Cross”.