Page 143 - ICSE Math 7

Alternatively,
By column method: Maths Info
2 – x + 3x 2 For convenience, arrange the terms of the polynomials in
2

– 6 + 4x + 2x ascending or descending order of powers of the variables.
2
– 12 + 6x – 18x 2 [Multiplying (2 – x + 3x ) by (–6)]
2
2
+ 8x – 4x + 12x 3 [Multiplying (2 – x + 3x ) by (4x)]
2
3
2
2
+ 4x – 2x + 6x 4 [Multiplying (2 – x + 3x ) by (2x )]
2
3
– 12 + 14x – 18x + 10x + 6x 4 (Adding the terms column wise)

Example 9: Simplify (3x – 4y) (x + 5) (2y + 2x)
Solution: (3x – 4y) (x + 5) (2y + 2x) = [(3x – 4y) (x + 5)] (2y + 2x)
= [3x(x + 5) – 4y(x + 5)] (2y + 2x)
2
= [3x + 15x – 4xy – 20y] (2y + 2x)
2
= 3x (2y + 2x) + 15x(2y + 2x) – 4xy(2y + 2x) – 20y(2y + 2x)
2
2
2
2
3
2
= 6x y + 6x + 30xy + 30x – 8xy – 8x y – 40y – 40xy
2
3
2
2
= 6x + 30x – 2x y – 8xy – 10xy – 40y 2
Division
1
For any variable x and positive integers m and n, x m = x m – n when m > n, and x m = n – m when m < n.
x n x n x
Division of a monomial by a monomial
Write the dividend in numerator and divisor in denominator. Simplify this fraction to get the result.
Example:
3
3 5
20x y 34ab c 2b 3–1 2b 2
4
2
2
3
3 5
(a) 20x y ÷ 4xy = = 5x 3–1 5–4 = 5x y (b) 34ab c ÷ 17abc = = =
y
4xy 4 17abc 2 c 2–1 c
Division of a polynomial by a monomial
Divide each term of the polynomial by the monomial and then add all the quotients obtained.
Example:
2
3ab 15a b 12ab 3 –1 5a
3
2
2
(a) (3ab + 15a b – 12ab ) ÷ (–3ab ) = + – = – + 4b
–3ab 2 –3ab 2 –3ab 2 b b
2 2
25xy 3 5x y 45y 5 9y 3
2 2
5
3
2
(b) (25xy – 5x y + 45y ) ÷ 5xy = – + = 5y – x +
5xy 2 5xy 2 5xy 2 x
Division of a polynomial by a polynomial
To divide a polynomial by a polynomial, follow the steps given below.
1. Write the dividend and the divisor in descending order of powers of the variables.
2. Divide the first term of the dividend by the first term of the divisor to get the first term of the quotient.
3. Multiply this term of the quotient by each term of the divisor and subtract the product from the dividend.
4. The remainder obtained after subtraction is 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.
5. Continue this process till the remainder is zero or a polynomial of degree lower than that of the
divisor.

129