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 $.

Given:

Given expression is $2x^3 + ax^2 + 2bx + 1$.

$x + 1$ is a factor of $2x^3 + ax^2 + 2bx + 1$ and $2a - 3b = 4$.

To do:

We have to find the values of $a$ and $b$.

Solution:

If $(x-m)$ is a root of $f(x)$ then $f(m)=0$.

This implies,

$(x+1)=x-(-1)$

Therefore,

$f(x)=2x^3 + ax^2 + 2bx + 1$

$f(-1)=0$

$\Rightarrow 2(-1)^3+a(-1)^2+2b(-1)+1=0$

$\Rightarrow -2+a-2b+1=0$

$\Rightarrow a-2b-1=0$

$\Rightarrow a=2b+1$....(i)

$2a - 3b = 4$    (Given)

Substituting equation (i) in $2a - 3b = 4$, we get,

$2(2b+1)-3b=4$

$4b+2-3b=4$

$b=4-2$

$b=2$

Substituting $b=2$ in equation (i), we get,

$a=2(2)+1$

$a=4+1$

$a=5$

The values of $a$ and $b$ are $5$ and $2$ respectively. 

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Updated on: 10-Oct-2022

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