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Let f(x) = ax 3 - 3x 2 + 4 and g(x) = 2x 3 - 5x + a By remainder theorem, f(2) = p ...(1) | x - 2 = 0 ⇒ x = 2 and g(2) = q ...(2) | x - 2 = 0 ⇒ x = 2 (1) gives a(2) 3 - 3(2) 2 + 4 = p ⇒ 8a - 8 = p ...(3) (2) gives 2(2) 3 - 5(2) + a = q ⇒ 6 + a = q ...(4) According to the question, p - 2q = 4 ⇒ (8a - 8) - 2 (6 + a) = 4 ⇒ 8a - 8 - 12 - 2a = 4 ⇒ 6a = 24 ⇒ a = 4

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Polynomials

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86. The polynomials ax³ - 3x² + 4 and 2x³ - 5x + a when divided by (x - 2) leave the remainders p and q respectively. If p - 2q = 4, find the value of a.

Let f(x) = ax³ - 3x² + 4
and    g(x) = 2x³ - 5x + a
By remainder theorem,
f(2) = p ...(1) | x - 2 = 0 ⇒ x = 2
and    g(2) = q ...(2) | x - 2 = 0 ⇒ x = 2
(1) gives
a(2)³ - 3(2)² + 4 = p
⇒    8a - 8 = p    ...(3)
(2) gives
2(2)³ - 5(2) + a = q
⇒    6 + a = q    ...(4)
According to the question,
p - 2q = 4
⇒ (8a - 8) - 2 (6 + a) = 4
⇒ 8a - 8 - 12 - 2a = 4
⇒    6a = 24
⇒    a = 4

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Test Yourself

Short Answer Type

86. The polynomials ax3 - 3x2 + 4 and 2x3 - 5x + a when divided by (x - 2) leave the remainders p and q respectively. If p - 2q = 4, find the value of a.

86. The polynomials ax³ - 3x² + 4 and 2x³ - 5x + a when divided by (x - 2) leave the remainders p and q respectively. If p - 2q = 4, find the value of a.