Maths Extension 1 Year 11 - Module 2 - Lesson 4

The Factor Theorem

Use this worksheet after reading the lesson to practise the key ideas and prove you can meet the success criteria.

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Date
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1. Lesson focus

Factor theorem. (x−a) is a factor of P(x) exactly when P(a)=0; after finding one factor, divide to factor the quotient.

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

1. Show that x−3 is a factor of x³−4x²+x+6, then factor completely.

2. Factor x³−3x²−4x+12 given that x=3 is a zero.

3. Factor x⁴−5x²+4 over the reals.

3. Challenge

If P(−2)=0, state a corresponding linear factor.

4. Multiple choice

If P(5)=0, which expression must divide P(x)?

A. x−5²

B. x+5

C. 5x−1

D. x−5

Worked answer key

Compare each step with your working. Keep stated intervals, domains and restrictions in your final answer.

1. Show that x−3 is a factor of x³−4x²+x+6, then factor completely.

P(3)=27−36+3+6=0. Division gives x²−x−2=(x−2)(x+1); full factorisation (x−3)(x−2)(x+1).

2. Factor x³−3x²−4x+12 given that x=3 is a zero.

Divide by x−3 to get x²−4=(x−2)(x+2); full factorisation (x−3)(x−2)(x+2).

3. Factor x⁴−5x²+4 over the reals.

Let u=x²: (u−1)(u−4)=(x−1)(x+1)(x−2)(x+2).

Challenge. If P(−2)=0, state a corresponding linear factor.

x−(−2)=x+2 is a factor.

Multiple choice. If P(5)=0, which expression must divide P(x)?

Correct answer: D. The factor theorem pairs the zero a with the factor x−a; for a=5 this is x−5.

HSCScience - Printable lesson worksheet - Maths Extension 1 Year 11 - Module 2 - Lesson 4

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