Factor and remainder theorems
Objective. Use evaluation to locate factors and unknown coefficients.
- 1. Learn
- 2. Worked example
- 3. Practice
- 4. Feedback
- 5. Continue
Learn: the key idea
The factor and remainder theorems link evaluating a polynomial at a number to its division properties. The remainder theorem states that dividing f(x) by (x−a) leaves remainder f(a); the factor theorem states that (x−a) is a factor exactly when f(a)=0. Example: for f(x)=x³−6x²+11x−6, since f(1)=1−6+11−6=0, (x−1) is a factor. This lets you test candidate roots quickly using the rational root list from the constant term's factors before dividing fully.
Worked example
Given f(x)=x³+2x²−5x−6, use the factor theorem to show (x+3) is a factor.
- 1Identify that (x+3)=(x−(−3)), so substitute x=−3 into f(x)
- 2Compute f(−3)=(−3)³+2(−3)²−5(−3)−6
- 3Evaluate: −27+18+15−6=0
- 4Since f(−3)=0, the factor theorem confirms (x+3) is a factor
Answer: f(−3)=0, so (x+3) is a factor
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Practice
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Start practisingCommon mistake. A common error is testing f(−a) instead of f(a) for the factor (x−a); for (x+3)=(x−(−3)), you must substitute x=−3, not x=3, into f(x).
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