Question

Assertion :If p(x) is a polynomial of degree $\ge$ 1 and ax+b is a factor of p(x) then we have $p(-\frac{b}{a})=0$. Reason: If p(x) is a polynomial of degree $\ge$ 1, then polynomial (x-a)(x-b) is a factor of p(x) if p(a)=0 and p(b)=0.

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Both Assertion and Reason are incorrect

Answer 1

The correct option is A Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

Let $p\left(x\right)$ be a polynomial of $degree\ge 1$.

$Assertion:$

If $ax+b$ is a factor of p(x), then :

$p\left(x\right)=(ax+b).q\left(x\right)$ where q(x) is another polynomial of degree$\ge$0.

$\Rightarrow p(-\frac{b}{a})=(a\times -\frac{b}{a}+b).q\left(x\right)=(-b+b).q\left(x\right)=0\times q\left(x\right)=0$

$\Rightarrow$ Assertion is correct.

$Reason:$

$(x-a)(x-b)$ is a factor of p(x), then :

$p\left(x\right)=(x-a)(x-b)q\left(x\right)$.

$\Rightarrow p\left(a\right)=(a-a)(a-b)q\left(x\right)=0andp\left(b\right)=(b-a)(b-b)q\left(x\right)=0$.

$\Rightarrow$ Reason is correct.

Reason explains assertion completely.

$\therefore$ The answer is $\left[A\right]$.