Question

If a and b are unequal and $x^2+ax+b$ and $x^2+bx+a$ have a common factor, then:

• A. a+b-1=0
• B. a+b+1=0
• C. a-b+1=0
• D. b-a+1=0

Answer 1

Answer: (B)

a+b+1=0

Let

$P\left(x\right)=x^2+ax+b$

$Q\left(x\right)=x^2+bx+a$

Given that $P\left(x\right)$ and $Q\left(x\right)$ have a common factor.

Let $(x-\alpha )$ be the common factor.

Therefore,

${\alpha }^2+a\alpha +b=0.....\left(1\right)$

${\alpha }^2+b\alpha +a=0.....\left(2\right)$

Subtracting ${eq}^n\left(1\right)$ from $\left(2\right)$, we have

$({\alpha }^2+a\alpha +b)-({\alpha }^2+b\alpha +a)=0$

${\alpha }^2+a\alpha +b-{\alpha }^2-b\alpha -a=0$

$(a-b)\alpha -(a-b)=0$

$(a-b)(\alpha -1)=0$

$\Rightarrow \alpha =1$

Substituting the value of $\alpha$ in ${eq}^n\left(1\right)$, we have

${\left(1\right)}^2+a\left(1\right)+b=0$

$\Rightarrow a+b+1=0$

Hence the correct answer is $\left(B\right)a+b+1=0$