Mathematics
When the polynomial x3 + 2x2 - 5ax - 7 is divided by (x - 1), the remainder is A and when the polynomial x3 + ax2 - 12x + 16 is divided by (x + 2), the remainder is B. Find the value of 'a' if 2A + B = 0.
Factorisation
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Answer
Given,
When the polynomial x3 + 2x2 - 5ax - 7 is divided by (x - 1), remainder is A.
x - 1 = 0 ⇒ x = 1
Substituting x = 1 in x3 + 2x2 - 5ax - 7 will give, remainder = A.
∴ (1)3 + 2(1)2 - 5a(1) - 7 = A
⇒ 1 + 2 - 5a - 7 = A
⇒ A = -(4 + 5a) ……(i)
Given,
When the polynomial x3 + ax2 - 12x + 16 is divided by (x + 2), remainder is B.
∴ x3 + ax2 - 12x + 16 = B
⇒ (-2)3 + a(-2)2 - 12(-2) + 16 = B
⇒ -8 + 4a + 24 + 16 = B
⇒ B = 32 + 4a ……(ii)
Given, 2A + B = 0
∴ -2(4 + 5a) + 32 + 4a = 0
⇒ -8 - 10a + 32 + 4a = 0
⇒ 24 - 6a = 0
⇒ 6a = 24
⇒ a = 4.
Hence, a = 4.
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