Question

If $x+y$ and $y+3x$ are two factor of the expression $\lambda x^3-\mu x^{2}y+x{y}^2+y^3$, then the third factor is

• A. $y+3x$
• B. $y-3x$
• C. $y-x$
• D. none of these

Answer 1

The correct option is B $y-3x$

Let $f\left(x\right)=\lambda x^3-\mu x^{2}y+x{y}^2+y^3$
As $x+y$ is factor
$\Rightarrow f(-y)=\lambda {(-y)}^3-\mu {(-y)}^{2}y+(-y)y^2+y^3\Rightarrow -y^3\lambda -\mu y^{2}y-y^3+y^3=0$
$\Rightarrow \lambda +\mu =0$ ...(1)
And as $y+3x$ is factor
$\Rightarrow f(-\frac{y}{3})=\lambda {(-\frac{y}{3})}^3-\mu {(-\frac{y}{3})}^{2}y+(-\frac{y}{3})y^2+y^3\Rightarrow -\frac{\lambda y^3}{27}-\frac{\mu y^3}{9}-\frac{y^3}{3}+y^3=0\Rightarrow -\lambda y^3-3\mu y^3-9y^3+27y^3=0$
$\Rightarrow \lambda +3\mu =18$ ...(2)
Solving (1) and (2), we get
$\lambda =-9,\mu =9$
Then
$f\left(x\right)=-9x^3-9x^{2}y+x{y}^2+y^3=(x+y)(y+3x)(y-3x)$
Hence, option 'B' is correct.