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 and or .
Notice, written in this form, x – k is a factor of . We can conclude if k is a zero of , then is a factor of .
Similarly, if is a factor of , then the remainder of the Division Algorithm 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 if and only if is a factor 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 .
- Confirm that the remainder is 0.
- Write the polynomial as the product of 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 is a factor of . Find the remaining factors. Use the factors to determine the zeros of the polynomial.
Solutions
We can use synthetic division to show that is a factor of the polynomial.
The remainder is zero, so 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 are –2, 3, and 5.
Try It 2
Use the Factor Theorem to find the zeros of given that is a factor of the polynomial.
Solution