Factor and Remainder Theorems

Explore remainder theorem, factor theorem, factoring after division.

Lesson 3 of 53 parts Practice Flashcards

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Remainder Theorem

Objective: Explain remainder theorem and use it to solve or interpret a problem.

The remainder on division by x−r equals p(r); evaluation can replace a full division when only the remainder is needed. Evaluate at x=r.

Worked examples

Try itFactor and Remainder Theorems — practice 1.1: Find the remainder when p(x)=x²+3x+7 is divided by x−5.Show answer

Step 1: Use the Remainder Theorem: remainder=p(5). Step 2: Compute 5²+3(5)+7. Step 3: The result is 47.

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Remainder Theorem

Objective: Explain remainder theorem and use it to solve or interpret a problem.

The remainder on division by x−r equals p(r); evaluation can replace a full division when only the remainder is needed. Evaluate at x=r.

Worked examples

Try itFactor and Remainder Theorems — practice 1.1: Find the remainder when p(x)=x²+3x+7 is divided by x−5.Show answer

Step 1: Use the Remainder Theorem: remainder=p(5). Step 2: Compute 5²+3(5)+7. Step 3: The result is 47.

Factor Theorem

Objective: Explain factor theorem and use it to solve or interpret a problem.

The linear expression x−r is a factor exactly when p(r)=0. A candidate zero is established by evaluation, not appearance alone. Evaluate p(r) and see whether it is nonzero.

Worked examples

Try itFactor and Remainder Theorems — practice 2.1: For p(x)=(x−8)(x+4), find p(8).Show answer

Step 1: The factor x−8 is zero at x=8. Step 2: The product is 0·(12). Step 3: Thus p(8)=0.

Factoring after Division

Objective: Explain factoring after division and use it to solve or interpret a problem.

Once a root is verified, divide out its linear factor and solve the lower-degree quotient; track repeated roots and original degree. The total multiplicity of factors should match the degree over complex numbers.

Worked examples

Try itFactor and Remainder Theorems — practice 3.1: Divide x²+13x+22 by x+2. What is the constant of the quotient?Show answer

Step 1: Factor the numerator as (x+2)(x+11). Step 2: Cancel x+2 in polynomial division. Step 3: The quotient is x+11; constant 11.

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