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

∑ DANEMATHICS

FREE CAPS MATHS RESOURCES

Grade 12 · Algebra

Polynomials: Remainder & Factor Theorem (Grade 12)

EXAM-STYLE CLASS TEST

Marks

Duration

55 minutes

Questions

Name:

Class:

Date:

Mark

/ 36

Instructions and Information

Answer ALL the questions in this question paper.
Answer QUESTION 1 by circling the letter (A–D) in the answer grid at the end of that section.
Show ALL calculations clearly.
Show all units where applicable.
Number the answers correctly according to the numbering system used in this question paper.
A non-programmable calculator may be used, unless stated otherwise.
Write neatly and legibly.

Question 1

[10 MARKS]

Four options are given as possible answers to the following questions. Choose the correct answer and circle the letter (A–D) in the grid at the end of this section. If you want to change your choice, put a cross through the wrong letter and circle your new choice.

1.1
By the Remainder Theorem, dividing f(x) by (x − a) leaves a remainder of …
(1)
A)f(a)
B)f(−a)
C)a
D)0
1.2
The remainder when x³ − 2x² + 5 is divided by (x − 2) is …
(1)
A)5
B)−11
C)−3
D)0
1.3
(x − 3) is a factor of f(x) exactly when …
(1)
A)f(3) = 0
B)f(−3) = 0
C)f(0) = 3
D)f(3) = 3
1.4
Which is a factor of f(x) = x³ − 4x² + x + 6?
(1)
A)(x − 2)
B)(x + 2)
C)(x − 1)
D)(x − 6)
1.5
Fully factorised, x³ − 4x² + x + 6 = …
(1)
A)(x − 2)(x − 3)(x + 1)
B)(x − 2)(x + 3)(x − 1)
C)(x + 2)(x − 3)(x + 1)
D)(x − 2)(x² − 2x − 3)
1.6
Solve for x: x³ − 4x² + x + 6 = 0
(1)
A)x = 2, x = 3 or x = −1
B)x = −2, x = −3 or x = 1
C)x = 2 only
D)x = 2, x = −3 or x = 1
1.7
A cubic polynomial has AT MOST … x-intercepts.
(1)
A)3
B)2
C)1
D)4
1.8
Dividing x³ + 2x² − 5x − 6 by (x + 1) gives a quotient of …
(1)
A)x² + x − 6
B)x² + 2x − 5
C)x² − x − 6
D)x² + 3x − 6
1.9
If f(x) = 2x³ + kx − 6 and (x − 1) is a factor, then k = …
(1)
A)4
B)−4
C)8
D)6
1.10
The remainder when f(x) is divided by (2x − 1) is …
(1)
A)f(½)
B)f(2)
C)f(−½)
D)f(1)

Answer grid — circle your answers for Question 1

1.1	A	B	C	D
1.2	A	B	C	D
1.3	A	B	C	D
1.4	A	B	C	D
1.5	A	B	C	D
1.6	A	B	C	D
1.7	A	B	C	D
1.8	A	B	C	D
1.9	A	B	C	D
1.10	A	B	C	D

Question 2

[13 MARKS]

Given f(x) = 2x³ − 5x² − 4x + 3.

2.1
Determine the remainder when f(x) is divided by (x − 1).
(3)
2.2
Show that (x − 3) is a factor of f(x).
(3)
2.3
Hence factorise f(x) COMPLETELY.
(4)
2.4
Hence solve for x: f(x) = 0, and write down the SUM of the three roots.
(3)

Question 3

[13 MARKS]

Given g(x) = x³ + px² − x − 6.

3.1
Determine the value of p for which (x + 1) is a factor of g(x).
(4)
3.2
Using this value of p, factorise g(x) completely.
(4)
3.3
Hence write down the x-intercepts of the graph of g, and calculate its y-intercept.
(3)
3.4
Explain why a cubic graph can have at most THREE x-intercepts.
(2)

TOTAL: 36 marks

This question paper consists of 3 questions.