Question

Divide $p\left(x\right)$ by $g\left(x\right)$ in the following case and verify division algorithm.
$p\left(x\right)=x^3+4x^2-5x+6;g\left(x\right)=x+1$

Answer 1

The division of $p\left(x\right)=x^3+4x^2-5x+6$ 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 $x^3+4x^2-5x+6$, the divisor is $x+1$, the quotient is $x^2+3x-8$ and the remainder is $14$, therefore,

To check:

$x^3+4x^2-5x+6=\left[\right(x+1\left)\right(x^2+3x-8\left)\right]+14\text{R.H.S}\Rightarrow \left[x\right(x^2+3x-8)+1(x^2+3x-8\left)\right]+14=x^3+3x^2-8x+x^2+3x-8+14=x^3+3x^2+x^2-8x+3x-8+14=x^3+4x^2-5x+6=\text{L.H.S}$

Hence, the division algorithm is verified.