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Question
Find the values of a and b when the polynomial f(x)= ax3 + 3x2 +bx -3 is exactly divisible by (2x+3) and leaves a remainder -3 when divided by (x+2).
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Solution
(2x +3) ⇒ x = `-3/2` .....(i)
(x + 2) ⇒ x = - 2 ...(ii)
putting (i) in polynomial , we get
`"f"(-3/2) = "a" xx (-3/2) xx (-3/2) xx (-3/2) + 3 xx (-3/2) xx (-3/2) + "b" xx (-3/2) - 3 = 0`
- 27 a + 54 - 12 b - 24 = 0
⇒ 27 a = -12 b + 30 ....(iii)
Putting (ii) in polynomial, and remainder is -3 we get
f(-2) = a × (-2) × (-2) × (-2) + 3 × (-2) × (-2) + b× (-2) - 3 = -3
b = 6 - 4a ..... (iv)
Combining (iii) and (iv), we get,
27a = -12 ×(6 - 4a) + 30
⇒ 27a= -72 + 48a + 30,
⇒ a=2, b= 6-4x2 = -2
a= 2, b= -2
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