Question

$f\left(x\right).g\left(x\right)=0$ where $f\left(x\right)$ and $g\left(x\right)$ are degree 2 polynomial with some of zeroes -2 and +2 respectivily.These constant term is same as well as coefficient of highest degree term. Then number of coincident zeroes of there product is

• A. 1
• B. 2
• C. 3
• D. None of these

Answer 1

The correct option is D None of these

$f\left(x\right)=a{x}^2+bx+cg\left(x\right)=a{x}^2+b^{\prime }x+c$

$f\left(x\right)g\left(x\right)=(a{x}^2+bx+c)(a{x}^2+b^{\prime }c+c)$

$0=(x^2+\frac{bx}{a}+\frac{c}{a})(x^2+\frac{b^{\prime }}{a}x+\frac{c}{a})$

$o=(x^2-(\alpha +\beta )x+\alpha \beta )(x^2-(\delta +\gamma )x+\delta \gamma )$

$o=(x-\alpha )(x-\beta )(x-\delta )(x-\gamma )$

$x=\alpha ,\beta ,\delta ,\gamma$

where $\alpha ,\beta$ are zeroes of $f\left(x\right)\delta \gamma$ are zeroes of $g\left(x\right)$

so all zeroes coincide with $f\left(x\right)g\left(x\right)$