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

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.

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

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:

We can factor the quadratic factor to write the polynomial as

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.