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Algebra II

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

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

There are two sets of remainder theorem worksheets

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

The Remainder Theorem is a fundamental concept in algebra that relates to polynomial functions and division. It states that when you divide a polynomial P(x) by a linear binomial of the form x − a, the remainder is equal to P(a).

In other words, if you have a polynomial P(x) and you want to find the remainder when it’s divided by x−a, you can substitute x = a into the polynomial P(x), and the result is the remainder of the division.

Here are the key steps to apply the Remainder Theorem:

- Identify the polynomial P(x) and the linear binomial x−a.
- Substitute the value of a into the polynomial P(x) and calculate P(a).
- Interpret the result:

If P(a)=0, then the remainder is 0, indicating that x−a is a factor of P(x).

If P(a) ≠ 0, then the remainder is P(a).

The Remainder Theorem is particularly useful when you want to check whether a particular binomial is a factor of a polynomial or when you need to find the remainder of a polynomial division.

Example:

Suppose you have the polynomial
P(x) = 2x^{3} − 5x^{2} + 3x − 1 and you want to find the remainder when P(x) is divided by x − 2.

- Identify the polynomial and the linear binomial.

Polynomial: P(x) = 2x^{3}− 5x^{2}+ 3x − 1

Linear binomial: x − 2 - Substitute x = 2 into P(x):

P(2) = 2(2)^{3}- 5(2)^{2}+ 3(2) − 1 = 16 − 20 + 6 − 1 = 1 - Interpret the result:

Since P(2)=1, the remainder when P(x) is divided by x−2 is 1.

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

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

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

**Printable**

(Answers on the second page.)

Remainder Theorem Worksheet #1 (cubic equations)

Remainder Theorem Worksheet #2 (quartic or quintic equations)

**Online**

Remainder Theorem

Remainder Theorem

Solve Cubic Equations

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