Maths Extension 1 Year 11 - Module 2 - Lesson 2
Polynomial Division
Name
Date
Class
1. Lesson focus
Polynomial division algorithm. Polynomial division writes P(x)=D(x)Q(x)+R(x), where deg R<deg D; synthetic division is efficient for linear divisors.
Success criteria
- Apply the lesson method accurately in a routine example.
- Show each algebraic or domain step and check the result.
- Explain why the method and restrictions fit the question.
2. Worked practice
2. Divide 2x³+x²−5x+3 by x+2.
3. Divide x⁴−1 by x²+1.
3. Challenge
Write the division algorithm form for x³+1 divided by x+1.
4. Multiple choice
When dividing by a linear polynomial, the remainder must be what degree or lower?
A. degree 1
B. A constant (degree 0)
C. degree 2
D. same degree as dividend
Worked answer key
Compare each step with your working. Keep stated intervals, domains and restrictions in your final answer.
1. Divide x³−2x²−5x+6 by x−3.
Synthetic division by 3 gives quotient x²+x−2 and remainder 0.
2. Divide 2x³+x²−5x+3 by x+2.
Division gives quotient 2x²−3x+1 and remainder 1, since (x+2)(2x²−3x+1)+1=2x³+x²−5x+3.
3. Divide x⁴−1 by x²+1.
x⁴−1=(x²+1)(x²−1), so quotient x²−1 and remainder 0.
Challenge. Write the division algorithm form for x³+1 divided by x+1.
x³+1=(x+1)(x²−x+1)+0.
Multiple choice. When dividing by a linear polynomial, the remainder must be what degree or lower?
Correct answer: B. The remainder degree is strictly less than the degree 1 divisor, so it is constant or zero.
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