Page 142 - ICSE Math 8

4. Subtract the following algebraic expressions.
2
2
2
2
2
(a) 7x – 3x – 5 from 8x + 4x – 8 (b) 4x – 7y from 7x + 3y 2
3
2
2
3
2
2
(c) (5x + 4y – 3z) – (3x – 2y – 5z) (d) (7x + 7xy – 6y ) – (3x + 5xy – 4y )
2
2
5. What should be added to 5x + 4xy to get 3x – 3xy?
6. If x = a + 2b and y = a – 2b, find (3x + 4y) – (5x – 7y) in terms of a and b.
3
4
3
4
7. What should be subtracted from 15a – 3a + 25a – 40 to get 3a + 10a – 12a – 20?
8. If X = 2p + 3q, Y = –5p + 4q and Z = –3p – 3q, find 3X – 2Y + Z.
Multiplication of Algebraic Expressions
n
m
If a is a variable and m, n are integers then, a × a = a m+n .
Multiplication of a monomial with a monomial
The steps for multiplying two monomials are as follows:
1. Find the product of the numerical coefficients and the literal coefficients separately.
2. Add the powers of the common variables and leave the other variables as it is.
3. Write the final product as the combination of numerals and literals.
Thus, product of monomials = (Product of the numerical coefficients) × (Product of the literal coefficients)
2
7 6 5
4 3 2
Example 7: Find the product of 3p qr, –4p q r and –12p q r .
2
4 3 2
7 6 5
4 3 2
7 6 5
2
Solution: 3p qr × (–4p q r ) × (–12p q r ) = {3 × (–4) × (–12)} × (p qr × p q r × p q r )
= (144)(p 2 + 4 + 7 × q 1 + 3 + 6 × r 1 + 2 + 5 )
13 10 8
= 144p q r
Multiplication of a polynomial and a monomial
Multiply each term of the polynomial by the given monomial and then add Try This
the products to get the answer. Find the products of:
2
2
2
2 2
Example 8: Find the product of 2a b – 3ab + 4ab – 6 and –5a b . (a) 2xy and –3x yz
2
3
(b) p – 2pq + 3q and 5p
2 2
2
2
Solution: (2a b – 3ab + 4ab – 6) × (–5a b )
2 2
2
2 2
2
= {2a b × (–5a b )} + {(–3ab ) × (–5a b )} + Maths Info
2 2
2 2
{4ab × (–5a b )} + {(–6) × (–5a b )}
2 2
4 3
3 3
3 4
= (–10a b ) + (15a b ) + (–20a b ) + (30a b ) The distributive laws of multiplication
over addition and subtraction are
3 4
3 3
2 2
4 3
= –10a b + 15a b – 20a b + 30a b followed here.
Multiplication of two polynomials
Multiply each term of one polynomial by each term of the other polynomial and then add the products to get
the answer. Alternatively, we can multiply polynomials by column method. For this, write one polynomial in
the first row and the other polynomial in the second row. Multiply each term of the polynomial in the first row
by each term of the polynomial in the second row. To get the result, add the products obtained by writing the
like terms in the same column.
Example 9: Find the product of 3x – 5y + 3 and –2x + 3y.
Solution: (3x – 5y + 3) × (–2x + 3y)
= {3x × (–2x)} + {(–5y) × (–2x)} + {3 × (–2x)} + (3x × 3y) + {(–5y) × 3y} + (3 × 3y)
2
2
= (–6x ) + (10xy) + (–6x) + (9xy) + (–15y ) + (9y)
2
2
= –6x + 19xy – 6x – 15y + 9y
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