Question

For the given statements select the correct option.

Assertion (A): If on dividing the polynomial $p\left(x\right)=x^2-3ax+3a-7$ by $(x+1)$, we get 6 as remainder, then $a=3$.

Reason (R): When a polynomial $p\left(x\right)$ is divided by $(x-a)$, then the remainder is $p\left(a\right)$.

Answer 1

We know that,
If on dividing the polynomial $p\left(x\right)$ by $(x-a)$, and if we get no remainder then $p\left(x\right)$ is perfectly divisible by $(x-a)$ and if we get $p\left(a\right)$ as remainder then $p\left(x\right)$ is not divisible by $(x-a)$

Now, given that on dividing $p\left(x\right)=x^2-3ax+3a-7$ by $(x+1)$, we get 6 as remainder.

i.e. $p(-1)=x^2-3ax+3a-7=6$

$\Rightarrow (-1{)}^2-3a(-1)+3a-7=6$

$\Rightarrow (-1{)}^2-3a(-1)+3a-7=6$

$\Rightarrow 6a-6=6$

$\Rightarrow a=2$

Therefore, assertion is false and reason is true.