# How do you multiply (3u^2 - n)^2?

Apr 10, 2015

To expand ${\left(3 {u}^{2} - n\right)}^{2}$ you can use the FOIL method. That stands for:

Fist
Outer
Inner
Last

In essence, you just want to multiply all combinations of the two brackets.

Since $\left(3 {u}^{2} - n\right)$ is squared, you can rewrite ${\left(3 {u}^{2} - n\right)}^{2} = \left(3 {u}^{2} - n\right) \cdot \left(3 {u}^{2} - n\right)$

Now you use the FOIL method to expand. "First" means you multiply the first term of the first bracket by the first term in the second bracket

$3 {u}^{2} \cdot 3 {u}^{2} = 9 {u}^{4}$

"Outer" means you multiply the first term in the first bracket by the last term in the second bracket- they are the outermost terms

$3 {u}^{2} \cdot - n = - 3 n {u}^{2}$

"Inner" means you multiply the second term of the first bracket with the first term in the second bracket

$- n \cdot 3 {u}^{2} = - 3 n {u}^{2}$

"Last" means you mutiply the last terms of each bracket

$- n \cdot - n = {n}^{2}$

Now you combine everything:

${\left(3 {u}^{2} - n\right)}^{2} = \left(3 {u}^{2} - n\right) \cdot \left(3 {u}^{2} - n\right) = 9 {u}^{4} - 3 n {u}^{2} - 3 n {u}^{2} + {n}^{2} = 9 {u}^{4} - 6 n {u}^{2} + {n}^{2}$