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\ 06-Jan-2025 Bharat Arora Proof-7 Reader’s Sign _______________________ Date __________

Column method

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We have, 4x – 2x + 5 Knowledge Desk
× 2x + 3 The first use of algebra was
done by Babylonians. The word
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8x – 4x + 10x (Multiplying by 2x) algebra comes from Arabic
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+ 12x – 6x + 15 (Multiplying by 3) word al-jabr which means
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8x + 8x + 4x + 15 (Adding the terms vertically) “reunion of broken parts”.
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Example 13: Multiply (3x – 2xy + 4y ) by (2x – 3y) and verify the result by taking x = 2 and y = –1.
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Solution: We have, (3x – 2xy + 4y ) × (2x – 3y)
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= (3x – 2xy + 4y ) × (2x) + (3x – 2xy + 4y ) × (–3y)
[Using distributive law]
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= 6x – 4x y + 8xy – 9x y + 6xy – 12y = 6x – 13x y + 14xy – 12y 3
Now we put x = 2 and y = –1 on both sides of the above result.
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LHS = (3x – 2xy + 4y ) × (2x – 3y)
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= [3(2) – 2(2)(–1) + 4(–1) ] × [2(2) – 3(–1)] = (12 + 4 + 4) × (4 + 3)
= 20 × 7 = 140
RHS = 6x – 13x y + 14xy – 12y 3
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= 6(2) – 13(2) (–1) + 14(2)(–1) – 12(–1) = 48 + 52 + 28 + 12 = 140
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\ LHS = RHS
Practice Time 8C
1. Find the product of the following.
(a) 3p (p + q) (b) 4.4ab(a – 2.2) (c) –16x(x + 2y) (d) x y (6x yz – 7xy z )
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2 2
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(e) –3x y(5y – xy + 3) (f) x ( 2 + y − 5z )
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7x
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2. Simplify –5n(mn + n ) + 278 and find its value for m = 3 and n = 4.
3. Multiply the following.
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(a) (4x – 7xy ) and (x – y ) (b)    3 x − 3   and    2 x + 2  
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y
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4 
3 
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(c) (p – 7) and (0.4p + 3) (d) (0.3p – 0.4q) and (2.5p – 5q)
4. Multiply the following.
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(a) (3x + 4x – 8) and (2x – 4x + 3) (b) (4x – 4y – 5) and (4x + 4y + 7)
5. Simplify.
(a) (3a – b )(a + 5b) + a b (b) (x + y)(2x + y) + (x + 2y)(x – y)
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(c) (x – 3x + 3)(5x – 2) – (4x + 5x – 6)(3x – 1) (d) 3a(4a – 5a) – a(a + 5) – 3a (7a – 6a + 5)
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6. (a) Add pq(p – 21p + 3) and 5pq( p + 4p – 7) (b) Subtract 5pq(p – q) from 3q(4p – 3pq + 8)
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7. Subtract b(b + b – 7) + 5 from 3b – 8 and find the value of expression obtained for b = –3.
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8. Simplify − aa( 3 − 4b + 5c) + , and find its value for a = 2, b = –2 and c = –3.
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Mathematics-8 202