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If (x+1) is a factor of 2x^(3)+ax^(2)+2b...

If (x+1) is a factor of `2x^(3)+ax^(2)+2bx+1`, then find the value of a and b given that 2a-3b=4.

Text Solution

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Let f(x) = `2x^(3) + ax^(2) + 2bx + 1`
`because (x + 1)` is a factor of f(x).
`therefore f(-1) = 0`
implies `2(-1)^(3) + a(-1)^(2) + 2b(-1) + 1 = 0`
implies -2 + a - 2b + 1 = 0
implies a - 2b = 1
implies a = 1 + 2b …(1)
Given that 2a - 3b = 4
implies 2(1 + 2b) - 3b = 4 [from (1)]
implies 2 + 4b - 3b = 4
implies b = 2
From equation (1)
a = 2b + 1
implies a = 2 `xx` 2 + 1 = 5
`therefore` a = 5, b = 2
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