Question

If $f\left(x\right)=g\left(x^3\right)+xh\left(x^3\right)$ is divisible by $x^2-x+1$ then which of the following is not true

• A. both $g\left(x\right)$ and $h\left(x\right)$ are divisible by $x+1$
• B. $g\left(x\right)$ is divisible by $x+1$
• C. $h\left(x\right)$ is divisible by $x+1$
• D. the statement $\left(A\right)$ is not true

Answer 1

Answer: (D)

the statement $\left(A\right)$ is not true

The roots of $x^2-x+1=0$ are $-w,-w^2$

Hence

$-w,-w^2$ are factors of $f\left(x\right)$

Therefore

$g(-w^3)-wh(-w^3)=0$

$g(-1)-wh(-1)=0...\left(i\right)$

Similarly

$g(-w^6)-w^{2}h(-w^6)=0$

$g(-1)-w^{2}h(-1)=0...\left(ii\right)$

Adding $\left(i\right)$ and $\left(ii\right)$, we get

$2g(-1)-h(-1)(w+w^2)=0$

$2g(-1)+h(-1)=0$

$h(-1)=-2g(-1)$

Substituting in $\left(i\right)$, we get

$g(-1)=0$

Hence

$x+1$ is factor of $g\left(x\right)$

Similarly $h(-1)=0$ and $x+1$ is a factor of $h\left(x\right)$