Page 26 - Pra U STPM 2022 Penggal 1 - Mathematics (T)

Mathematics Term 1 STPM Chapter 1 Functions

2
4
5
For example, if P(x) = 2x – x – 5x – 2x – 3 and Q(x) = x + x + 1, then P(x) may be found using long
2
division as shown below. Q(x)
2x – 3x + x – 3 1
3
2
3
5
2
3
4
2
x + x + 1 2x – x + 0x – 5x – 2x – 3 Add a term 0x to avoid
confusion during working.
2x + 2x + 2x 3
5
4
–3x – 2x – 5x 2
4
3
–3x – 3x – 3x 2
4
3
x – 2x – 2x
2
3
x + x + x
3
2
2
– 3x – 3x – 3
2
– 3x – 3x – 3
0
Hence, P(x) = 2x – 3x + x – 3, i.e. a polynomial of degree 3.
3
2
Q(x)
If it is known that the division of two polynomials is exact, the quotient may also be obtained by using the
method as shown in Example 16 below.
Example 16
2
3
Find the quotient if x – 4x + 5x – 2 can be divided by (x – 2) exactly.
Solution: Let the quotient be the polynomial q(x).
3
2
x – 4x + 5x – 2
Hence, ––––––––––––––– = q(x)
x – 2
3
2
i.e. x – 4x + 5x – 2 ≡ q(x) · (x – 2) Multiply both sides by (x – 2)
1442443

Polynomial of degree 3 Polynomial of degree 2
Since q(x) is a polynomial of degree 2 (i.e. a quadratic function), it must be of
2
the form ax + bx + c.
2
2
3
Hence, x – 4x + 5x – 2 ≡ (ax + bx + c)(x – 2)
2
2
3
= ax + bx + cx – 2ax – 2bx – 2c
2
3
= ax + (b – 2a)x + (c – 2b)x – 2c
3
Equating coefficients of x : 1 = a
2

Equating coefficients of x : –4 = b – 2a

–4 = b – 2(1)
b = –2
Equating coefficients of x: 5 = c – 2b
5 = c – 2(–2)
c = 1
2
Hence, a = 1, b = –2 and c = 1, and the quotient is x – 2x + 1.
23
01a STPM Math T T1.indd 23 3/28/18 4:20 PM