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All courses Math III · A-APR.2 2 of 55
Apply the Remainder Theorem to connect p(a), remainders, and factors x-a.

Evaluate a polynomial at a given value to find the remainder

Problem
Evaluate the polynomial to find the remainder: p(x) = x³ - 2x + 5 at x = 2.
Your answer

Hint

Substitute x = 2 into p(x) = x³ - 2x + 5. The remainder is the value of the polynomial at that x-value.

Compute each term separately: 2³, -2(2), and +5. Then add the results.

Solution walkthrough

01

Substitute x = 2

\[p(2)~=~2^3~-~2(2)~+~5\]

To find the remainder, evaluate the polynomial at x = 2 by replacing each x with 2.

02

Compute each term

\[\begin{aligned} 2^3~=~8 \\ -2(2)~=~-4 \\ +5~=~5 \end{aligned}\]

Evaluate the power term, the linear term, and the constant term separately to avoid sign mistakes.

03

Add the values

\[8~-~4~+~5~=~9\]

Now combine the three evaluated terms to get the value of the polynomial.

04

State the remainder

\[\text{Remainder}~=~9\]

Since p(2) = 9, the remainder is 9.

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Another way

  1. You can think of this as using the Remainder Theorem: the remainder when dividing by x - 2 is p(2).

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Common mistake

A common mistake is to compute 2³ correctly as 8 but forget that -2x becomes -2(2) = -4, not +4.