# 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

If *k* is a zero, then the remainder *r* is

Notice, written in this form, *x* – *k* is a factor of

*k*is a zero of

Similarly, if

*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

### 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 $\left(x-k\right)$.
- Confirm that the remainder is 0.
- Write the polynomial as the product of $\left(x-k\right)$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

**polynomial**.

### Solutions

We can use synthetic division to show that

The remainder is zero, so

We can factor the quadratic factor to write the polynomial as

By the Factor Theorem, the zeros of

### Try It 2

Use the Factor Theorem to find the zeros of

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