Polynomials: Remainder & Factor Theorem (Grade 12) — Worksheet & Memo

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Grade 12 · Algebra

Polynomials: Remainder & Factor Theorem (Grade 12)

MEMORANDUM

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/ 22

Answer all questions. Show all your working — marks are awarded for method as well as the final answer. Teacher copy: accept any correct equivalent method.

1
Determine the remainder when f(x) = x³ − 4x² + x + 6 is divided by (x − 1).
(2)
f(1) = 1 − 4 + 1 + 6 = 4✓✓ (2)
2
Show that (x − 2) is a factor of f(x) = x³ − 4x² + x + 6.
(2)
f(2) = 8 − 16 + 2 + 6 = 0, so by the factor theorem (x − 2) is a factor.✓✓ (2)
3
Factorise f(x) = x³ − 4x² + x + 6 completely.
(3)
(x − 2)(x − 3)(x + 1)✓✓✓ (3)
4
Solve for x: x³ − 4x² + x + 6 = 0
(2)
x = 2, x = 3 or x = −1✓✓ (2)
5
Determine the remainder when f(x) = 2x³ + x − 3 is divided by (x + 1).
(2)
f(−1) = −2 − 1 − 3 = −6✓✓ (2)
6
If (x − 3) is a factor of f(x), what is f(3)?
(1)
0✓ (1)
7
Which whole-number values should you test first when factorising a monic cubic with integer coefficients (just x³ in front) and constant term 12?
(2)
The factors of 12: ±1, ±2, ±3, ±4, ±6, ±12✓✓ (2)
8
Factorise f(x) = 2x³ + x² − 13x + 6 completely.
(4)
f(2) = 16 + 4 − 26 + 6 = 0, so (x − 2) is a factor; 2x³ + x² − 13x + 6 = (x − 2)(2x² + 5x − 3) = (x − 2)(2x − 1)(x + 3)✓✓✓✓ (4)
9
Solve for x: x³ − 7x + 6 = 0
(4)
f(1) = 0 → (x − 1)(x² + x − 6) = (x − 1)(x + 3)(x − 2) → x = 1, 2 or −3✓✓✓✓ (4)

TOTAL: 22 marks