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Question
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.
Solution
Let f(x) = 2x3 + ax2 + bx - 14 ...(1)
as (x - 2) is factor of (1)
Put x - 2 = 0
⇒ x = 2 in (1)
f(2) = 2(2)3 + a(2)2 + b(2) - 14
0 = 16 + 4a + 2b - 14
or
4a + 2b = -2
or 2a + b = -1 ...(2)
Again when f(x) is divided by (x - 3), it leaves remainder 52
Put x - 3 = 0
⇒ x = 3
f(3) = 2(3)3 + a(3)2 + b(3) - 14
52 = 54 + 9a + 3b - 14
52 = 9a + 3b + 40
52 - 40 = 9a + 3b
⇒ 12 = 9a + 3b
or
4 = 3a + b ...(3)
Solving (2) and (3)
3a + b = 4
2a + b = -1
Sub - - +
a = 5
Substitute a = 5 in 3a + b = 4
⇒ 3 x 5 + b = 4
15 + b = 4
⇒ b = 4 - 15
b = -11.
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