How do you multiply (8v ^ { 2} - 5v + 3) ( 2v ^ { 2} + 5v - 2)?

May 27, 2017

See a solution process below:

Explanation:

To multiply these two terms you multiply each individual term in the left parenthesis by each individual term in the right parenthesis.

$\left(\textcolor{red}{8 {v}^{2}} - \textcolor{red}{5 v} + \textcolor{red}{3}\right) \left(\textcolor{b l u e}{2 {v}^{2}} + \textcolor{b l u e}{5 v} - \textcolor{b l u e}{2}\right)$ becomes:

$\left(\textcolor{red}{8 {v}^{2}} \times \textcolor{b l u e}{2 {v}^{2}}\right) + \left(\textcolor{red}{8 {v}^{2}} \times \textcolor{b l u e}{5 v}\right) - \left(\textcolor{red}{8 {v}^{2}} \times \textcolor{b l u e}{2}\right) - \left(\textcolor{red}{5 v} \times \textcolor{b l u e}{2 {v}^{2}}\right) - \left(\textcolor{red}{5 v} \times \textcolor{b l u e}{5 v}\right) + \left(\textcolor{red}{5 v} \times \textcolor{b l u e}{2}\right) + \left(\textcolor{red}{3} \times \textcolor{b l u e}{2 {v}^{2}}\right) + \left(\textcolor{red}{3} \times \textcolor{b l u e}{5 v}\right) - \left(\textcolor{red}{3} \times \textcolor{b l u e}{2}\right)$

$16 {v}^{4} + 40 {v}^{3} - 16 {v}^{2} - 10 {v}^{3} - 25 {v}^{2} + 10 v + 6 {v}^{2} + 15 v - 6$

We can now group and combine like terms:

$16 {v}^{4} + 40 {v}^{3} - 10 {v}^{3} - 16 {v}^{2} - 25 {v}^{2} + 6 {v}^{2} + 10 v + 15 v - 6$

$16 {v}^{4} + \left(40 - 10\right) {v}^{3} + \left(- 16 - 25 + 6\right) {v}^{2} + \left(10 + 15\right) v - 6$

$16 {v}^{4} + 30 {v}^{3} + \left(- 35\right) {v}^{2} + 25 v - 6$

$16 {v}^{4} + 30 {v}^{3} - 35 {v}^{2} + 25 v - 6$