Question

Find the quotient when the polynomial $x^3+x^2+1$ is divided by $x+2$ using synthetic division method.

• A. $2x^2-3x+2$
• B. $x^2-x+2$
• C. $4x^2-x+2$
• D. $x^2-x+6$

Answer 1

The correct option is B $x^2-x+2$

Index form of the dividend polynomial is :

$x^3+x^2+1=x^3+x^2+0x+1$

Coefficient form of the dividend polynomial is :

(1,1,0,1)

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

$\begin{array}{ccccc}& x^3& x^2& x^1& x^0\\ & 1& 1& 0& 1\\ -2& \downarrow & & & \\ & 1& & & \end{array}$

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 1. Addition of 1 and -2 which is -1, is to be written in the third row and second column.

Similarly by multiplying and adding, we have

$\begin{array}{ccccc}& x^3& x^2& x^1& x^0\\ & 1& 1& 0& 1\\ -2& \downarrow & -2& 2& -4\\ & 1& -1& 2& |--3\end{array}$

The last addition is the remainder, which is ' -3 '.

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

So, quotient is $x^2-x+2$, remainder is '-3'