Question

Find the quotient and remainder when $3x^7-x^6+31x^4+21x+5$ is divided by $x+2$.

Answer 1

Let us divide $3x^7-x^6+31x^4+21x+5$ by $x+2$ using synthetic division method

The divisor must be in the form $x-c$

Thus, we write it as $x-(-2)$

Hence we have to divide the co-efficient of equation by $-2$ as follows:

$\begin{array}{cccccccc}3& -1& 0& 31& 0& 0& 21& 5\\ & -6& 14& -28& -6& 12& -24& 6\end{array}$
__________________________________
$\begin{array}{cccccccc}3& -7& 14& 3& -6& 12& -3& 11\end{array}$

The numbers in the last row represent the coefficients of the quotient and the remainder. The degree of the first term of the quotient is one less than that of the dividend. As the degree of the dividend is $7,$ the degree of the quotient is $6.$

Hence the quotient is $3x^6-7x^5+14x^4+3x^3-6x^2+12x-3$

And the remainder is $11$.