How do you use the binomial theorem to expand and simplify the expression (y-4)^3?

Jun 22, 2018

We can expand this expression by remembering the rule for expanding cubic functions.

Explanation:

You could rewrite ${\left(y - 4\right)}^{3}$ as $\left(y - 4\right) \left(y - 4\right) \left(y - 4\right)$ or as $\left(y - 4\right) \left({y}^{2} - 8 y + 16\right)$, but that would take ages to solve! Instead, let's think of it this way:

${\left(a - b\right)}^{3} = {a}^{3} - 3 {a}^{2} b + 3 a {b}^{2} - {b}^{3}$

Using this, we can make our equation look a lot less scary:

${\left(y - 4\right)}^{3} = {y}^{3} - 3 \left({y}^{2}\right) \left(4\right) + 3 \left(y\right) \left({4}^{2}\right) - \left({4}^{3}\right)$
${\left(y - 4\right)}^{3} = {y}^{3} - 12 {y}^{2} + 48 y - 64$

Since no like terms exist, our equation is completely simplified.