Page 32 - Pra U STPM 2022 Penggal 1 - Mathematics (T)

Mathematics Term 1 STPM Chapter 1 Functions

Example 24

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Given that (x – 2) is a factor of f(x), where f(x) ≡ ax – 10x + bx – 2, a, b  R. 1
When f(x) is divided by (x – 3), its remainder is 16. Find the values of a and b.
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Solution: Given that (x – 2) is a factor of f(x) ≡ ax – 10x + bx – 2.
By the factor theorem,
f(2) = 0
a(2) – 10(2) + b(2) – 2 = 0
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8a – 40 + 2b – 2 = 0
8a + 2b = 42
4a + b = 21 ………… 
Given that when f(x) is divided by (x – 3), the remainder is 16.
By the remainder theorem,
f(3) = 16
a(3) – 10(3) + b(3) – 2 = 16
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27a – 90 + 3b – 2 = 16
27a + 3b = 108 ………… 
9a + b = 36
 – : 5a = 15
a = 3

Substituting a = 3 into ,
12 + b = 21
b = 9
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Hence, f(x) = 3x – 10x + 9x – 2.
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Exercise 1.5

1. Use the remainder theorem to find the remainder when each of the following polynomials is divided by
the linear polynomial given.
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(a) x + 4x – 3x + 2; x – 2 (b) x – 3x + 2x – 1; x + 2
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(c) x – x + 6; x + 1 (d) x + 2x + 1; 2x – 1
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(e) x – 2x + x + 1; 2x – 3 (f) x + 15x – 1; 3x + 1
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2. Determine whether each of the following linear polynomials is a factor of the polynomial given.
(a) x + 2; x + 4x + 4x 2 (b) x – 1; x + 3x – 6x + 3
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(c) x + 1; x – 2x + 6x + 9 (d) x – 2; x – 4x + 3x + 2
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(e) 2x + 1; 2x – 3x + 2x + 2 (f) 2x – 1; x + 4x – 3x – 1
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3. Factorise each of the following polynomials.
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(a) 2x – 3x + 1 (b) 3x – 2x – 7x – 2
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(c) x – x – 72 (d) x + x + x
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(e) 4x – 13x + 6 (f) 4x – 4x – 9x + x + 2
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4. If (x – 2) is a factor of ax + 3x – 2x + a, find the value of a.
5. Show that (x – a) is a factor of x + (1 – a)x + (3 – a)x – 3a.
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