The Factor Theorem is another theorem that helps us analyze polynomial equations. It tells us how the zeros of a polynomial are related to the factors.
the factor theorem
According to the factor theorem, [latex]k[/latex] is a zero of [latex]f(x)[/latex] if and only if [latex](x−k)[/latex] is a factor of [latex]f(x)[/latex].
Let’s walk through the proof of the theorem.
Recall that the Division Algorithm.
[latex]f(x) = (x - k)q(x) + r[/latex]
If [latex]k[/latex] is a zero, then the remainder [latex]r[/latex] is [latex]f(k) = 0[/latex] and [latex]f(x) = (x - k)q(x) + 0[/latex] or [latex]f(x) = (x - k)q(x)[/latex].
Notice, written in this form, [latex]x - k[/latex] is a factor of [latex]f(x)[/latex]. We can conclude if [latex]k[/latex] is a zero of [latex]f(x)[/latex], then [latex]x - k[/latex] is a factor of [latex]f(x)[/latex].
Similarly, if [latex]x - k[/latex] is a factor of [latex]f(x)[/latex], then the remainder of the Division Algorithm [latex]f(x) = (x - k)q(x) + r[/latex] is [latex]0[/latex]. This tells us that [latex]k[/latex] is a zero.
This pair of implications is the Factor Theorem. As we will soon see, a polynomial of degree [latex]n[/latex] in the complex number system will have [latex]n[/latex] zeros. We can use the Factor Theorem to completely factor a polynomial into the product of [latex]n[/latex] factors. Once the polynomial has been completely factored, we can easily determine the zeros of the polynomial.
How to: Given a factor and a third-degree polynomial, use the Factor Theorem to factor the polynomial.
Use synthetic division to divide the polynomial by [latex](x-k)[/latex]
Confirm that the remainder is [latex]0[/latex].
Write the polynomial as the product of [latex](x-k)[/latex] and the quadratic quotient.
If possible, factor the quadratic.
Write the polynomial as the product of factors.
Show that [latex](x + 2)[/latex] is a factor of [latex]x^3 - 6x^2 - x + 30[/latex]. Find the remaining factors. Use the factors to determine the zeros of the polynomial.
We can use synthetic division to show that [latex](x + 2)[/latex] is a factor of the polynomial.
The remainder is zero, so [latex](x + 2)[/latex] is a factor of the polynomial. We can use the Division Algorithm to write the polynomial as the product of the divisor and the quotient:
[latex](x + 2)(x^2 - 8x + 15)[/latex]
We can factor the quadratic factor to write the polynomial as
[latex](x + 2)(x - 3)(x - 5)[/latex]
By the Factor Theorem, the zeros of [latex]x^3 - 6x^2 - x + 30[/latex] are [latex]-2, 3,[/latex] and [latex]5[/latex].
Find the zeros of [latex]f\left(x\right)=4{x}^{3}-3x - 1[/latex] and graph the function.
The Rational Zero Theorem tells us that if [latex]\frac{p}{q}[/latex] is a zero of [latex]f\left(x\right)[/latex], then p is a factor of –1 and q is a factor of 4.
[latex]\begin{array}{l}\frac{p}{q}=\frac{\text{Factors of the constant term}}{\text{Factors of the leading coefficient}}\hfill \\ \text{}\frac{p}{q}=\frac{\text{Factors of -1}}{\text{Factors of 4}}\hfill \end{array}[/latex]
The factors of –1 are [latex]\pm 1[/latex] and the factors of 4 are [latex]\pm 1,\pm 2[/latex], and [latex]\pm 4[/latex]. The possible values for [latex]\frac{p}{q}[/latex] are [latex]\pm 1,\pm \frac{1}{2}[/latex], and [latex]\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].
Dividing by [latex]\left(x - 1\right)[/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]\left(x - 1\right)\left(4{x}^{2}+4x+1\right)[/latex].
The quadratic is a perfect square. [latex]f\left(x\right)[/latex] can be written as [latex]\left(x - 1\right){\left(2x+1\right)}^{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].
