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Question
Polynomials bx2 + x + 5 and bx3 − 2x + 5 are divided by polynomial x - 3 and the remainders are m and n respectively. If m − n = 0 then find the value of b.
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Solution
Let p(x) = bx2 + x + 5 and q(x) = bx3 - 2x + 5.
The remainder when p(x) = bx2 + x + 5 is divided by (x − 3) is m.
By remainder theorem,
Remainder = p(3) = m
∴ b × (3)2 + 3 + 5 = m
⇒ m = 9b + 8 ...(1)
The remainder when q(x) = bx3 − 2x + 5 is divided by (x - 3) is n.
By remainder theorem,
Remainder = q(3) = n
∴ b × (3)3 − 2 × 3 + 5 = n
⇒ n = 27b − 6 + 5
⇒ n = 27b − 1 ...(2)
Now,
m − n = 0
⇒ (9b + 8) - (27b − 1) = 0 [Using (1) and (2)]
⇒ 9b − 27b + 8 + 1 = 0
⇒ −18b + 9 = 0
⇒ −18b = −9
`⇒ b = (−9)/(−18) = 1/2`
Thus, the value of b is `1/2`.
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