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Question
Find the values of a and b in the polynomial f(x) = 2x3 + ax2 + bx + 10, if it is exactly divisible by (x+2) and (2x-1).
Solution
(x+2) ⇒ x = -2 .... (i)
(2x - 1) ⇒ x = `1/2` .......(ii)
Putting (i) in polynomial, we get
f(-2) = 2 × (-2) × (-2) × (-2) + a × (-2) × ( -2) + b × (-2) + 10 = 0
⇒ - 16 + 4a - 2b + 10 = 0
⇒ a = `"b"/2 + 3/2` .....(iii)
Putting (ii) in polynomial, we get
`"f"(1/2) = 2 xx (1/2) xx (1/2) xx (1/2) + "a" xx (1/2) xx (1/2) + "b" xx (1/2) + 10 = 0`
`=> 1/4 + "a"/4 + "b"/2 + 10 = 0`
⇒ a = - 2b - 41 ......(iv)
Combining (iii) and (iv), we get,
`"b"/2 + 3/2 = "a" = -2"b" - 41`
`=> ("b + 3")/2 = - 2 "b" - 41`
⇒ b + 3 = - 4b - 82
⇒ 5b = - 85
⇒ b = -17
and a = -7
⇒ a = - 7 , b = -17
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