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All courses Algebra II · A-APR.2 9 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
For \(p(x)~=~x^3~-~2x~+~5\), evaluate \(p(2)\) to find the remainder.
Your answer
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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

Connect evaluation to the remainder

\[\text{remainder}=p(a)\]

For division by x-a, the Remainder Theorem says the remainder is p(a). The prompt directly asks for p(2), so a=2 and the required remainder is the polynomial's value at 2.

02

Substitute into every x term

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

Replace both occurrences of x by 2. The linear term -2x becomes -2(2), while the constant +5 is unchanged.

03

Evaluate with signs intact

\[2^3-2(2)+5=8-4+5=9\]

The cubic term is 8 and the linear term is -4. Adding 8-4+5 gives 9.

04

Check and state the remainder

\[\text{remainder}:~9\]

A quick recomputation groups 8+5=13 and then subtracts 4, again giving 9. Therefore p(2)=9, so the requested remainder is 9, choice A.

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Another way

  1. Use synthetic division with 2 and coefficients 1, 0, -2, 5; the final bottom entry is 9.

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Common mistake

Do not replace -2x by -2. Substituting x=2 makes that entire term -2(2)=-4.

Solution walkthrough video