Question

The polynomial $p\left(x\right)=x^4-2x^3+3x^2-ax+3a-7$ when divided by $x+1$ leaves the remainder $19$.

Find the values of $a$. Also find the remainder when $p\left(x\right)$ is divided by$x+2$.

Answer 1

Step 1. Find the value of $a$.

It is given that

$p\left(x\right)=x^4-2x^3+3x^2-ax+3a-7$ when divided by $x+1$ leaves the remainder $19$

So, $p\left(-1\right)=19$

$$\begin{gathered} p(-1)={\left(-1\right)}^4-2{\left(-1\right)}^3+3{\left(-1\right)}^2-a\left(-1\right)+3a-7 \\ \Rightarrow 19=1+2+3+a+3a-7 \\ \Rightarrow 19=4a-1 \\ \Rightarrow 4a=20 \\ \therefore a=5 \end{gathered}$$

Step 2. Find the remainder when $p\left(x\right)$ is divided by$x+2$.

Put $a=5,x=-2$ in the polynomial $p\left(x\right)=x^4-2x^3+3x^2-ax+3a-7$

$$\begin{gathered} p(-2)={\left(-2\right)}^4-2{\left(-2\right)}^3+3{\left(-2\right)}^2-5\left(-2\right)+3\times 5-7 \\ =16+16+12+10+15-7 \\ =62 \end{gathered}$$

Hence, the value of $a=5$ and the remainder is $62$.