Question

Use factor theorem and find the value of k for which (x + 2) is a factor of $f\left(x\right)=(x+1{)}^7+(2x+k{)}^3$.

Answer 1

Given that,

$(x+2)$ is a factor of $f\left(x\right)=(x+1{)}^7+(2x+k{)}^3$

According to the factor theorem, if $(x-a)$ is a factor of $f\left(x\right)$, then $f\left(a\right)=0$

$\therefore f(-2)=0$

$\Rightarrow (-2+1{)}^7+(2(-2)+k{)}^3=0$ [Putting $x=-2$ in $f\left(x\right)$]

$\Rightarrow (-1{)}^7+(k-4{)}^3=0$

$\Rightarrow -1+(k-4{)}^3=0$

$\Rightarrow (k-4{)}^3=1$

$\Rightarrow k-4=1$

$\therefore k=5$

Hence, the value of $k$ for which $(x+2)$ is a factor of $f\left(x\right)=(x+1{)}^7+(2x+k{)}^3$ is $5$