Use the Factor Theorem to solve a polynomial equation

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. Recall that the Division Algorithm tells us

[latex]f\left(x\right)=\left(x-k\right)q\left(x\right)+r\\[/latex].

If k is a zero, then the remainder r is [latex]f\left(k\right)=0\\[/latex] and [latex]f\left(x\right)=\left(x-k\right)q\left(x\right)+0\\[/latex] or [latex]f\left(x\right)=\left(x-k\right)q\left(x\right)\\[/latex].

Notice, written in this form, x – k is a factor of [latex]f\left(x\right)\\[/latex]. We can conclude if k is a zero of [latex]f\left(x\right)\\[/latex], then [latex]x-k\\[/latex] is a factor of [latex]f\left(x\right)\\[/latex].

Similarly, if [latex]x-k\\[/latex] is a factor of [latex]f\left(x\right)\\[/latex], then the remainder of the Division Algorithm [latex]f\left(x\right)=\left(x-k\right)q\left(x\right)+r\\[/latex] is 0. This tells us that k is a zero.

This pair of implications is the Factor Theorem. As we will soon see, a polynomial of degree n in the complex number system will have n zeros. We can use the Factor Theorem to completely factor a polynomial into the product of n factors. Once the polynomial has been completely factored, we can easily determine the zeros of the polynomial.

A General Note: The Factor Theorem

According to the Factor Theorem, k is a zero of [latex]f\left(x\right)\\[/latex] if and only if [latex]\left(x-k\right)\\[/latex] is a factor of [latex]f\left(x\right)\\[/latex].

How To: Given a factor and a third-degree polynomial, use the Factor Theorem to factor the polynomial.

  1. Use synthetic division to divide the polynomial by [latex]\left(x-k\right)\\[/latex].
  2. Confirm that the remainder is 0.
  3. Write the polynomial as the product of [latex]\left(x-k\right)\\[/latex] and the quadratic quotient.
  4. If possible, factor the quadratic.
  5. Write the polynomial as the product of factors.

Example 2: Using the Factor Theorem to Solve a Polynomial Equation

Show that [latex]\left(x+2\right)\\[/latex] is a factor of [latex]{x}^{3}-6{x}^{2}-x+30\\[/latex]. Find the remaining factors. Use the factors to determine the zeros of the polynomial.

Solutions

We can use synthetic division to show that [latex]\left(x+2\right)\\[/latex] is a factor of the polynomial.

Synthetic division with divisor -2 and quotient {1, 6, -1, 30}. Solution is {1, -8, 15, 0}

The remainder is zero, so [latex]\left(x+2\right)\\[/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]\left(x+2\right)\left({x}^{2}-8x+15\right)\\[/latex]

We can factor the quadratic factor to write the polynomial as

[latex]\left(x+2\right)\left(x - 3\right)\left(x - 5\right)\\[/latex]

By the Factor Theorem, the zeros of [latex]{x}^{3}-6{x}^{2}-x+30\\[/latex] are –2, 3, and 5.

Try It 2

Use the Factor Theorem to find the zeros of [latex]f\left(x\right)={x}^{3}+4{x}^{2}-4x - 16\\[/latex] given that [latex]\left(x - 2\right)\\[/latex] is a factor of the polynomial.

Solution