Page 118 - Start Up Mathematics_8 (Non CCE)

Example 3: Divide:

2 2 3
2 3 2
3 2 2
2 2 2
(a) 8(x y z + x y z + x y z ) by 4x y z (NCERT)
1
3
2
4
3 4
2
2
(b) (9ab – 15a b ) by (–3ab ) (c) (2x + 3x – x ) by x
3
2 2
2 3
2
3 2
3 2 2
2
2
Solution: (a) 8(x y z + x y z + x y z ) ÷ 4x y z
3
2 2
32
2 3
8(xy z + x yz + xy z )
2
2
=
2 2
2
4xy z
32
2
2
3
8 Ê xy z + x yz + xy z ˆ Ê xy z xy z xy z ˆ
2 2
2 3
3 2
2 2
2 3
2
3
2
= ¥ Á ˜ = 2 Á + + 2 2 ˜
2
2 2
2 2
2
2
2
22
4 Ë xy z ¯ Ë xy z x yyz xy z ¯
32-
= 2 x ( 32- + y 32- + z ) = 2 (x + y + z)
2
3 4
34
2
2
3 4
(b) (9ab – 15a b ) ÷ (–3ab ) = 9ab - 15ab = 9ab 2 - Ê 15ab ˆ
2 ˜
- 3ab 2 - 3ab 2 Á Ë - 3ab ¯
2 2
2 2
= –3 – (–5 × a 3 – 1 × b 4 – 2 ) = –3 – (–5a b ) = –3 + 5a b
4
3
1
2
3
4
(c) (2x + 3x – x ) ÷ x = 2x + 3x - x 2 = 2x 4 + 3x 3 - x 2
3 1 x 1 x 1 x 1 x
3 3 3 3
Ê
= 2 ∏ 1ˆ ¥ x x 4 + Ê Á Ë 3∏ 1ˆ ¥ x x 3 - Ê Á Ë 1∏ 1ˆ ¥ x x 2
Á
˜
˜
˜
3¯
3¯
3¯
Ë
= (2 × 3) × x 4 – 1 – (3 × 3) × x 3 – 1 – (1 × 3) × x 2 – 1
2
3
= 6x + 9x – 3x
EXERCISE 6.3
Divide:
8
2
2
5
4
3
(a) x y – 3xy by y (b) y – 8y + 5y by y 2 (c) –4x + xy – 12xz by –4x
3
2
6
4
2
3
4
3
2
2
3
4
(d) x – 3x + 1 x by 3x (e) –p + 2p + 4p + 2p by 2p (f) 2x + 3 2x + 2x - 6x by 2x
2
4 3
5 4
2 2
3 2 2
3 2
2 3 2
(g) 4x y + 10x y – 12x y by (–2x y ) (h) 36x y z – 28x y z by (–4xyz)
Division of a Polynomial by a Binomial Using Long Division Method
Step 1: The polynomial is the dividend and the binomial is the divisor. Arrange the terms in them in the
descending order of their degrees.
Step 2: Divide the first term of the dividend by the first term of the divisor to get the first term of the quotient.
Multiply the divisor by the first term of the quotient and subtract the result from the dividend to get
the remainder.
Step 3: Taking the remainder, if any, as dividend, repeat step II to obtain the second term of the quotient.
Step 4: Continue the procedure till the remainder is zero or a polynomial of degree less than that of the
divisor.
110