Question

If $(x+1)$ is a factor of $2x^3+a{x}^2+2bx+1$, then find the value $(a+b)$, given that $2a-3b=4$

Answer 1

It is given that $(x+1)$ is a factor of $p\left(x\right)=2x^3+a{x}^2+2bx+1$ which means that $-1$ is a zero of $p\left(x\right)$

Substituting the value $x=-1$ in the given $p\left(x\right)$, we get:

$p(-1)=2(-1{)}^3+a(-1{)}^2+2b(-1)+1=(2\times -1)+(a\times 1)-2b+1=-2+a-2b+1=a-2b-1\Rightarrow a-2b=1.......\left(1\right)$

Also it is given that $2a-3b=4$ $....\left(2\right)$

Multiply equation 1 by $2$:

$2(a-2b)=2\times 1\Rightarrow 2a-4b=2.......\left(3\right)$

Subtract equation 3 from equation 2 as follows:

$(2a-3b)-(2a-4b)=4-2\Rightarrow 2a-2a-3b+4b=2\Rightarrow b=2$

Substituting the value of $b$ in equation 1 we get:

$a-(2\times 2)=1\Rightarrow a-4=1\Rightarrow a=1+4=5$

Therefore, $a=5$ and $b=2$ and $a+b=5+2=7$

Hence, $(a+b)=7$.