Question

Divide $p\left(x\right)$ by $g\left(x\right)$ in the following case and verify division algorithm.
$p\left(x\right)=x^3-1;g\left(x\right)=x-1$

Answer 1

The division of $p\left(x\right)=x^3-1$ by $g\left(x\right)=x-1$ is as shown above:

Now we know that the division algorithm states that:

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

Here, the dividend is $x^3-1$, the divisor is $x-1$, the quotient is $x^2+x+1$ and the remainder is $0$, therefore,

$x^3-1=\left[\right(x-1\left)\right(x^2+x+1\left)\right]+0\Rightarrow x^3-1=\left[x\right(x^2+x+1)-1(x^2+x+1\left)\right]+0\Rightarrow x^3-1=x^3+x^2+x-x^2-x-1+0\Rightarrow x^3-1=x^3+x^2-x^2+x-x-1\Rightarrow x^3-1=x^3-1$

Hence, the division algorithm is verified.