There are three sets of factor theorem and remainder theorem worksheets:

Examples, solutions, videos, and worksheets to help Algebra II students learn about the factor theorem and the remainder theorem.

### How to use the Factor Theorem to determine whether a binomial is a factor of the given polynomial?

There are two sets of factor theorem worksheets

- Factor theorem with cubic equations
- Factor theorem with quartic or quintic equations

To determine whether a given binomial is a factor of a polynomial, you can use the Factor Theorem. The Factor Theorem states that if a polynomial P(x) has a factor of the form x−a, then P(a)=0. In other words, if you substitute the value a into the polynomial and the result is zero, then (x−a) is a factor of the polynomial.

Here are the steps to determine whether a binomial is a factor:

- Identify the binomial and the polynomial.
- Use the Factor Theorem. Substitute the value from the binomial (let’s call it a) into the polynomial, and calculate P(a).
- Interpret the result:

If P(a) = 0, then the binomial (x−a) is a factor of the polynomial.

If P(a) ≠ 0, then the binomial is not a factor of the polynomial.

Example:

Determine whether x+3 is a factor of the polynomial

x^{3} + 3x^{2} + 3x + 9

- Identify the binomial and the polynomial.
Binomial: x+3
Polynomial: x
^{3} + 3x^{2} + 3x + 9
- Use the Factor Theorem. Substitute a=-3 into P(x):
P(3) = (-3)
^{3} + 3(-3)^{2} + 3(-3) + 9
- Interpret the result:
P(3) = -27 + 27 - 9 + 9 = 0
Since P(3)=0, this means that x+3 is a factor of the polynomial P(x).

Have a look at this video if you need to review how to use the factor theorem.

Click on the following worksheet to get a printable pdf document.

Scroll down the page for more **Factor Theorem Worksheets**.

### More Factor Theorem Worksheets

**Printable**

(Answers on the second page.)

Factor Theorem Worksheet #1 (cubic equations)

Factor Theorem Worksheet #2 (quartic or quintic equations)

**Online**

Remainder Theorem

Factor Theorem

Solve Cubic Equations

More Printable Worksheets

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