Mathematics
Evaluate the following products without multiplying directly:
(i) 103 × 107
(ii) 95 × 96
(iii) 104 × 96
Polynomials
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Answer
(i) 103 × 107
We can write it as,
= (100 + 3)(100 + 7)
[∵ (x + a)(x + b) = x2 + (a + b)x + ab]
Putting x = 100, a = 3, b = 7
= (100)2 + (3 + 7)(100) + (3)(7)
= 10000 + (10)(100) + 21
= 10000 + 1000 + 21
= 11021
Hence, 103 × 107 = 11021
(ii) 95 × 96
We can write it as,
[100 + (-5)][100 + (-4)]
[∵ (x + a)(x + b) = x2 + (a + b)x + ab]
Putting x = 100, a = -5 , b = -4
= (100)2 - [(-5) + (-4)](100) + (-5) x (-4)
= 10000 - (9)(100) + 20
= 10000 - 900 + 20
= 10020 - 900
= 9120
Hence, 95 × 96 = 9120
(iii) 104 × 96
We can write it as,
(100 + 4)(100 - 4)
[∵ (a + b)(a - b) = a2 - b2]
Putting a = 100, b = 4
= (100)2 - (4)2
= 10000 - 16
= 9984
Hence, 104 × 96 = 9984
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