Page 27 - Pra U STPM 2022 Penggal 1 - Mathematics (T)

Mathematics Term 1 STPM Chapter 1 Functions

Example 17

1 Find the constants A, B and C, such that
2
x – 5x + 12 ≡ A (x – 1)(x – 2) + B(x + 1)(x – 1) + C(x – 2)(x + 3).
Solution: Since the given equation is an identity, we can substitute suitable values of x into
the identity to determine the values of A, B and C.
When x = 1, 1 – 5(1) + 12 = A(0) + B(0) – 4C

8 = –4C
C = –2
2
When x = 2, 2 – 5(2) + 12 = A(0) + 3B + C(0)
6 = 3B
B = 2
When x = 0, 12 = 2A – B – 6C
12 = 2A – 2 – 6(–2)
12 = 2A – 2 + 12
A = 1
Hence, A = 1, B = 2 and C = –2.

2
Note: We can also find the constants A, B and C in Example 17 by equating the coefficients of x , x and the
constants on both sides of the identity.

Exercise 1.4

1. If P(x) = 2x – 3x + 4x + 1, find the values of
2
3
1
(a) P(0) (b) P 1 2 (c) P(2)
2
2. If Q(x) = 2x – 5x + x – 3, find the values of
4
2
3
(a) Q(–1) (b) Q 1 2 (c) Q(2)
2
3. If F(x) = x + x – 1 and G(x) = 1 + 2x, find
2
(a) F(x) + 2G(x) (b) F(x) – G(x) (c) 3F(x) + x · G(x)
(d) (1 + x) · F(x) (e) G(x) · F(x)
4. By substituting suitable values of x into each of the identities below, find the values of the constants A, B
and C.
(a) 3x + 3 ≡ A(x – 1) + B(2 + x)
(b) 7x + 6 ≡ A(x – 2) + B(x + 3)
(c) 2x + 5 ≡ A(x + 1) + B(x – 2)
(d) 2x – 5x + 7 ≡ A(x + 1)(x – 2) + B(x + 1)(x – 1) + C(x – 2)(x – 1)
2
2
2
(e) x – 6x – 19 ≡ A(x + 5)(x – 1) + B(x – 1) + C(x + 5)(x + 1)
5. Find the product of
2
3
2
(a) 2x – x + 7 and x + 2 (b) 3x – 2x + 5x – 1 and 2x – 3
2
(c) x + 3x – 2 and 4x – x + 1 (d) 5x – 2x + 3 and x – 1
2
3
2
3
4
(e) 3x – 2x + 6x – 4 and 3x + 2
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