Question

Find the quotient and remainder on dividing $p\left(x\right)$ by $g\left(x\right)$ in the following case, without actual division.

$p\left(x\right)=x^2+7x+10$ and $g\left(x\right)=x-2$

Answer 1

Step 1: State the given data and use the remainder theorem

It is given that, $p\left(x\right)=x^2+7x+10$

And, $g\left(x\right)=x-2$

Here, $p\left(x\right)$ is the dividend and $g\left(x\right)$ is the divisor.

According to the remainder theorem, if a polynomial $p\left(x\right)$ is divided by a binomial $\left(x-a\right)$, the remainder obtained is $p\left(a\right)$.

Now, on equating $g\left(x\right)$ to zero, we get,

$g\left(x\right)=0$

$\Rightarrow x-2=0$

$\Rightarrow$ $x=2$

Step 2: Use synthetic division

So, using the remainder theorem,

So, the quotient $=1\times x+9=x+9$

And, the remainder $=28$

Hence, the required quotient and remainder are $\left(x+9\right)$ and $28$ respectively.