Question

Divide (6 + 19x + x² − 6x³) by (2+ 5x − 3x²) and verify the division algorithm.

Answer 1

$\begin{array}{l}\text{First, we will write the given polynomials in their standard forms in decreasing order of}\\ \mathrm{degrees\; and\; then\; perform\; the\; division\; as\; shown\; below:}\end{array}$



$\begin{array}{l}\\ \therefore \text{Quotient}=(2x+3)\mathrm{and}\mathrm{r}\text{emainder}=0\\ \Rightarrow (\text{Quotient}\times \mathrm{d}\text{ivisor})+\text{remainder =}(2x+3)\times (2+5x-3x^2)+0\\ =4x+6+10x^2+15x-6x^3-9x^2+0\\ =-6x^3+x^2+19x+6\\ =\text{dividend}\\ \text{Thus,}(\mathrm{q}\text{uotient}\times \text{divisor})+\text{remainder}=\mathrm{dividend}\\ \text{Hence, the division algorithm is verified}\text{.}\\ \end{array}$