Page 143 - ICSE Math 8

Alternatively, by column method:
3x – 5y + 3
–2x + 3y
2
–6x + 10xy – 6x [Multiplying (3x – 5y + 3) by (–2x)]
2
9xy –15y + 9y [Multiplying (3x – 5y + 3) by (3y)]
2
2
–6x + 19xy – 6x – 15y + 9y (Adding the terms column wise)
Example 10: Simplify (x + 5)(x – 3)(3x + 2).
Solution: (x + 5)(x – 3)(3x + 2) = [(x + 5)(x – 3)](3x + 2) Try These
= [x(x – 3) + 5(x – 3)](3x + 2) 1. Multiply (4x + 3a) and
2
= [x – 3x + 5x – 15](3x + 2) (3x – 2a).
2
2
2
= [x + 2x – 15](3x + 2) 2. Multiply 2x + 3xy + 3y by
2
2
2z – 2xy + y .
2
= x (3x + 2) + 2x(3x + 2) – 15(3x + 2)
2
3
2
= 3x + 2x + 6x + 4x – 45x – 30
3
2
= 3x + 8x – 41x – 30
Division of Algebraic Expressions
m
n
For any variable a and positive integers m and n, a ÷ a = a m – n
Division of a monomial by a monomial
To divide two monomials, first divide the numerical coefficients and then divide the literal coefficients.
Thus, quotient of two monomials = (Quotient of numerical coefficients) × (Quotient of literal coefficients)
For example, 3 5
3
3 5
2 2
y
(a) 27x y ÷ 9xy = 27 × x y = 3x 3–1 5–3 = 3x y
9 xy 3
3 2 4
r
3 2 4
2 4
(b) 11p q r ÷ 15p q = 11 × p q r = 11p 3–2 4 = 11pr 4
2 4
15 p q 15q 4–2 15q 2
Division of a polynomial by a monomial
Divide each term of the polynomial by the monomial and then add all the quotients obtained. For example,
2
16a b 4ab 2 12b 3 4 1 3b
2
3
2 2
2
(a) (16a b + 4ab – 12b ) ÷ (4a b ) = + – = + – 2
2 2
4a b 4a b 4a b b a a
2 2
2 2
2 3
2 2
5 4
3 2
5 4
2 3
(b) (25x y – 5xy – x y ) ÷ (–5x y ) = 25x y – 5xy – x y = –5y + 1 + x y
3 2
2
3 2
3 2
–5x y –5x y –5x y x x y 5
Division of a polynomial by a polynomial
To divide a polynomial by a polynomial, follow the given steps.
Step 1: Arrange the terms of the dividend and the divisor in descending order of the powers of the variables.
Insert zeros in place of the missing terms.
Step 2: Divide the first term of the dividend by the first term of the divisor to get the first term of the quotient.
Step 3: Multiply each term of the divisor by the first term of the quotient and subtract the product from the
dividend.
Step 4: The remainder obtained after subtraction along with the remaining terms of the dividend forms the
new dividend. Divide the first term of the new dividend by the first term of the divisor and write the
result of division as the second term of the quotient.
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