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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