Question

If the polynomials $f\left(x\right)=x^2+5x-p$ and $g\left(x\right)=x^2-2x+6p$ have a common factor then the common factor is _____

• A. $x+2$
• B. $x$
• C. $x+4$
• D. Either (2) or (3)

Answer 1

Answer: (D)

Either (2) or (3)

Given,$f\left(x\right)=x^2+5x-p$

$g\left(x\right)=x^2-2x+6p$

Let ${}^{\prime }{\alpha }^{\prime }$ be common root.

$f\left(\alpha \right)={\alpha }^2+5\alpha -p=0$

$g\left(\alpha \right)={\alpha }^2-2\alpha +6p=0$

$f\left(\alpha \right)-g\left(\alpha \right)=7\alpha -7p=0$

$\alpha =p$

$f\left(p\right)=p^2+5p-p=0$

$p^2+4p=0$

$p=0,-4$

$p=0\Rightarrow f\left(x\right)=x^2+4x=x(x+4)$ roots are $0,-4$

$\Rightarrow g\left(x\right)=x^2-2x=x(x-2)$ roots are $0,2$

$p=0\Rightarrow$ common root is $x=0$

$p=-4\Rightarrow f\left(x\right)=x^2+5x+4=(x+1)(x+4)$

$x=-1,-4$

$\Rightarrow g\left(x\right)=x^2-2x=24=(x-6)(x+4)$

$x=6,-4$

$p=-4\Rightarrow$ commom root is $x=-4$

$\therefore$ common factor is $xor(x+4)$