Factor and Remainder Theorems

Develop remainder theorem, factor theorem, factoring after division.

Lesson 3 of 43 parts Practice Flashcards

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Remainder Theorem

Objective: Explain remainder theorem, apply the rule, and justify relevant restrictions.

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. Identify the stated domain before using the rule, preserve restrictions during algebraic simplification, and interpret the result in the units or graph features requested. Use the worked example to connect the symbolic rule with a concrete calculation.

Formulas

$p(x)=(x−c)q(x)+p(c)$

Worked examples

Try itFind the remainder when p(x)=x²+3x+5 is divided by x−7.Show answer

Step 1: Use the Remainder Theorem: remainder=p(7). Step 2: Compute 7²+3(7)+5. Step 3: The result is 75.

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Remainder Theorem

Objective: Explain remainder theorem, apply the rule, and justify relevant restrictions.

Formulas

$p(x)=(x−c)q(x)+p(c)$

Worked examples

Try itFind the remainder when p(x)=x²+3x+5 is divided by x−7.Show answer

Step 1: Use the Remainder Theorem: remainder=p(7). Step 2: Compute 7²+3(7)+5. Step 3: The result is 75.

Factor Theorem

Objective: Explain factor theorem, apply the rule, and justify relevant restrictions.

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. Identify the stated domain before using the rule, preserve restrictions during algebraic simplification, and interpret the result in the units or graph features requested. Use the worked example to connect the symbolic rule with a concrete calculation.

Formulas

p(c)=0 ⇔ (x−c) is a factor of p(x)

Worked examples

Try itFor p(x)=(x−7)(x+3), find p(7).Show answer

Step 1: The factor x−7 is zero at x=7. Step 2: The product is 0·(10). Step 3: Thus p(7)=0.

Factoring after Division

Objective: Explain factoring after division, apply the rule, and justify relevant restrictions.

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. Identify the stated domain before using the rule, preserve restrictions during algebraic simplification, and interpret the result in the units or graph features requested. Use the worked example to connect the symbolic rule with a concrete calculation.

Formulas

$p(x)=d(x)q(x)+r(x)$

Worked examples

Try itDivide x²+10x+21 by x+3. What is the constant of the quotient?Show answer

Step 1: Factor the numerator as (x+3)(x+7). Step 2: Cancel x+3 in polynomial division. Step 3: The quotient is x+7; constant 7.

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