Question

If $(x+k)$ is the HCF of $a{x}^2+x+b$ and $x^2+cx+d$, then what is the value of $k$?

• A. $\frac{b+d}{a+c}$
• B. $\frac{a+b}{c+d}$
• C. $\frac{a-b}{c-d}$
• D. NONE OF THE ABOVE

Answer 1

The correct option is D NONE OF THE ABOVE

Given $X+R$ is the HCF of $a{x}^2+x+band{x}^2+cx+d$

then $X+R$ is divisor of $a{x}^2+x+b$ and $x^2+cx+d$

but $x=-k$ in the given function

then $a{x}^2+x+b=a(-k{)}^2+(-k)+b=a{k}^2-k+b=\mathrm{0...}(1)$

and $x^2+cx+d=(-k{)}^2+c(-k)+d=k^2-ck+d=0...(2)$

Solving (1) and (2)

$K^2=\frac{k-b}{a}=ck-d$

$\Rightarrow k-b=ack-ad$

$\Rightarrow k(1-ac)=b-ad$

$\Rightarrow K=\frac{b-ad}{1-ac}$