If x+1 is a factor of $2 x^{3} + a x^{2} + 2 b x + 1$ , then find the values of a and b given that $2 a - 3 b = 4$ .

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Solution

Since, $x + 1$ is a factor of $p (x) = 2 x^{3} + a x^{2} + 2 b x + 1$
Then, by factor theorem, $p (- 1) = 0$
$\Rightarrow - 2 + a - 2 b + 1 = 0 \Rightarrow a - 2 b = 1$ ...(i)
Also, $2 a - 3 b = 4$ ...(ii)
On multiplying (i) by 2 and (ii) by 1, we get
$2 a - 4 b = 2$
$2 - a - + 3 b = 4 - - ---------- -$
$- b = - 2$
$∴ b = 2$
On putting $b = 2$ in (i), we get
$a - 2 \times 2 = 1 \Rightarrow a = 5$
$∴ a = 5, b = 2$

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