Question

Divide $p\left(x\right)$ by $g\left(x\right)$ in the following case and verify division algorithm.
$p\left(x\right)=x^4-3x^2-4;g\left(x\right)=x+2$

Answer 1

The division of $p\left(x\right)=x^4-3x^2-4$ by $g\left(x\right)=x+2$ is as shown above:

Now we know that the division algorithm states that:

Dividend$=($Divisor$\times$quotient$)+$Remainder

Here, the dividend is $x^4-3x^2-4$, the divisor is $x+2$, the quotient is $x^3-2x^2+x-2$ and the remainder is $0$, therefore,

$x^4-3x^2-4=\left[\right(x+2\left)\right(x^3-2x^2+x-2\left)\right]+0\Rightarrow x^4-3x^2-4=\left[x\right(x^3-2x^2+x-2)+2(x^3-2x^2+x-2\left)\right]+0\Rightarrow x^4-3x^2-4=x^4-2x^3+x^2-2x+2x^3-4x^2+2x-4+0\Rightarrow x^4-3x^2-4=x^4-2x^3+2x^3+x^2-4x^2-2x+2x-4\Rightarrow x^4-3x^2-4=x^4-3x^2-4$

Hence, the division algorithm is verified.