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Question
(x – 2) is a factor of the expression x3 + ax2 + bx + 6. When this expression is divided by (x – 3), it leaves the remainder 3. Find the values of a and b.
Solution
As x – 2 is a factor of
f(x) = x3 + ax2 + bx + 6
∴ f(2) = 0
∴ (2)3 + a(2)2 + b(2) + 6 = 0
⇒ 8 + 4a + 2b + 6 = 0
⇒ 4a + 2b = –14
⇒ 2a + b = –7 ....(i)
as on dividing f(x) by x – 3
remainder = 3
∴ f(3) = 3
∴ (3)3 + a(3)2 + b(3) + 6 = 3
⇒ 27 + 9a + 3b + 6 = 3
⇒ 9a + 3b = –30
⇒ 3a + b = –10 ....(ii)
Solving simultaneously equation (i) and (ii),
∴ 2a + b = –7
3a + b – 10
Subtracting – – +
–a = 3
a = –3
Subtracting value of a in equation (i)
2(–3) + b = –7
∴ –6 + b = –7
∴ b = –1
∴ a = –3, b = –1.
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