ProofHard3 marks
Using the factor theorem, show that x - 2 is a factor of x³ + x² - 4x - 4. Hence, factorise the polynomial completely.
Topic: Remainder and factor theorem
Previous-year question practice database
Answer
Exam answer
− 4x − 4 = (x − 2)(x + 1)(x + 2)
Explanation
f(x) = x3 + x2 − 4x − 4 f(2) = 23 + 22 − 4 × 2 − 4 = 8 + 4 − 8 − 4 = 0 ∴ x − 2 is a factor of f(x)
x − 2 | x3
+ x2
− 4x − 4 | x2
+ 3x + 2 x3 − 2x2 3x2 − 4x 3x2 − 6x 2x − 4 2x − 4 x x2 + 3x + 2 = (x + 1)(x + 2) ∴ x3 + x2 − 4x − 4 = (x − 2)(x + 1)(x + 2)
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