Question

Find the quotient when the polynomial $x^2+1$ is divided by $x+2$ using synthetic division method.

Answer 1

Index form of the dividend polynomial is :

$x^2+1=x^2+0x+1$

Coefficient form of the dividend polynomial is :

(1,0,1)

Divisor polynomial =x+2

Steps:
1. Draw the horizontal and vertical lines as below :



2. Divisor is x+2. Take opposite number of 2 which is -2.Write -2 to the left of the vertical line. Write the coefficient form of the dividend polynomial in the first row and write the first coefficient as it is in the third row

$\begin{array}{ccccc}& x^2& x^1& x^0& \\ & 1& 0& 1& \\ -2& \downarrow & & & \\ & 1& & & \end{array}$

3. The product of 1 in the third row
with divisor -2 is -2. Write this -2 in the second row and second column below the coefficient 0. Addition of 0 and -2 which is -2, is to be written in the third row and second column.

Similarly by multiplying and adding, we have

$\begin{array}{cccc}& x^2& x^1& x^0\\ & 1& 0& 1\\ -2& \downarrow & -2& 4\\ & 1& -2& |-5\end{array}$

The last addition is the remainder, which is ' 5 '.

The coefficient form of the quotient is (1, -2).

So, quotient is $x-2$, remainder is ' 5 '