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प्रश्न
If (x – 2) is a factor of the expression 2x3 + ax2 + bx – 14 and when the expression is divided by (x – 3), it leaves a remainder 52, find the values of a and b.
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उत्तर
Let p(x) = 2x3 + ax2 + bx - 14
Given, (x – 2) is a factor of p(x),
⇒ Remainder = p(2) = 0
⇒ 2(2)3 + a(2)2 + b(2) – 14 = 0
⇒ 16 + 4a + 2b – 14 = 0
⇒ 4a + 2b + 2 = 0
⇒ 2a + b + 1 = 0 ...(1)
Given, when p(x) is divided by (x – 3), it leaves a remainder 52
∴ p(3) = 52
∴ 2(3)3 + a(3)2 + b(3) – 14 = 52
⇒ 54 + 9a + 3b - 14 - 52 = 0
⇒ 9a + 3b – 12 = 0
⇒ 3a + b – 4 = 0 ...(2)
Subtracting (1) from (2), we get,
a – 5 = 0 ⇒ a = 5
From (1),
10 + b + 1 = 0 ⇒ b = –11
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