Polynomial division
Objective. Divide polynomials and use quotient-remainder structure.
- 1. Learn
- 2. Worked example
- 3. Practice
- 4. Feedback
- 5. Continue
Learn: the key idea
Polynomial division splits one polynomial by another to give a quotient and remainder, just as 17÷5 gives 3 remainder 2. For f(x)÷(x−a) the result is f(x)=(x−a)q(x)+r, where r is a constant found by the remainder theorem or by algebraic long division matching coefficients term by term. Example: dividing x³−2x²−5x+6 by (x−1) gives quotient x²−x−6 with remainder 0, since 1−2−5+6=0. Always align powers of x in order, subtracting carefully at each stage, and check by expanding the quotient back out.
Worked example
Divide x³+2x²−5x−6 by (x−2) and state the quotient and remainder.
- 1Set up long division with x³+2x²−5x−6 over (x−2)
- 2x³÷x=x², multiply (x−2) by x² giving x³−2x², subtract to leave 4x²−5x−6
- 34x²÷x=4x, multiply (x−2) by 4x giving 4x²−8x, subtract to leave 3x−6
- 43x÷x=3, multiply (x−2) by 3 giving 3x−6, subtract to leave remainder 0
Answer: Quotient x²+4x+3, remainder 0
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Practice
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Start practisingCommon mistake. Students forget to include a zero coefficient for a missing power, e.g. treating x³−5x+6 as having no x² term in the layout at all, which misaligns subtraction; always write 0x² as a placeholder column.
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