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Question
The polynomials 2x3 – 7x2 + ax – 6 and x3 – 8x2 + (2a + 1)x – 16 leaves the same remainder when divided by x – 2. Find the value of ‘a’.
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Solution
Let f(x) = 2x3 – 7x2 + ax – 6
x – 2 = 0 `\implies` x = 2
When f(x) is divided by (x – 2), remainder = f(2)
∴ f(2) = 2(2)3 – 7(2)2 + a(2) – 6
= 16 – 28 + 2a – 6
= 2a – 18
Let g(x) = x3 – 8x2 + (2a + 1)x – 16
When g(x) is divided by (x – 2), remainder = g(2)
∴ g(2) = (2)3 – 8(2)2 + (2a + 1)(2) – 16
= 8 – 32 + 4a + 2 – 16
= 4a – 38
By the given condition, we have:
f(2) = g(2)
2a – 18 = 4a – 38
4a – 2a = 38 – 18
2a = 20
a = 10
Thus, the value of a is 10.
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