Maths Extension 1 Year 11 - Module 2 - Lesson 10
Reducible Polynomials
Name
Date
Class
1. Lesson focus
Reducibility over the real numbers. A polynomial is reducible over the reals if it factors into lower-degree real polynomials; use substitutions such as u=x² for even powers.
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. Factor x⁴−1 into real factors.
3. Factor x⁴+5x²+4 over the reals.
3. Challenge
Show that x³−x has three distinct real linear factors.
4. Multiple choice
Which is a real factorisation of x⁴−1?
A. (x²+1)²
B. (x−1)(x+1)(x²+1)
C. (x−1)⁴
D. x²−1
Worked answer key
Compare each step with your working. Keep stated intervals, domains and restrictions in your final answer.
1. Factor x⁴−5x²+4 over the reals.
Set u=x²: (u−1)(u−4), so (x−1)(x+1)(x−2)(x+2).
2. Factor x⁴−1 into real factors.
(x²−1)(x²+1)=(x−1)(x+1)(x²+1); x²+1 is irreducible over the reals.
3. Factor x⁴+5x²+4 over the reals.
(x²+1)(x²+4); both quadratic factors have no real roots, so this is a real factorisation.
Challenge. Show that x³−x has three distinct real linear factors.
x³−x=x(x²−1)=x(x−1)(x+1).
Multiple choice. Which is a real factorisation of x⁴−1?
Correct answer: B. Difference of squares gives (x²−1)(x²+1), then x²−1=(x−1)(x+1).
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