How do you use the binomial theorem to expand and simplify the expression (5-3y)^3?

Oct 29, 2017

$125 - 225 y + 135 {y}^{2} - 27 {y}^{3}$

Explanation:

$\text{using the "color(blue)"binomial theorem}$

•color(white)(x)(a+b)^n=sum_(r=0)^n""^nC_r a^(n-r)b^r

$\text{where the binomial coefficient }$

""^nC_r=(n!)/(r!(n-r)!)

$\text{we can obtain the coefficients using the appropriate row}$
$\text{of "color(blue)"Pascal's triangle ""for n = 3}$

$\Rightarrow \textcolor{red}{1} \textcolor{w h i t e}{x} \textcolor{red}{3} \textcolor{w h i t e}{x} \textcolor{red}{3} \textcolor{w h i t e}{x} \textcolor{red}{1}$

$\text{here "a=5" and } b = - 3 y$

$\Rightarrow {\left(5 - 3 y\right)}^{3}$

$= \textcolor{red}{1} {.5}^{3} {\left(- 3 y\right)}^{0} + \textcolor{red}{3} {.5}^{2} {\left(- 3 y\right)}^{1} + \textcolor{red}{3} {.5}^{1} {\left(- 3 y\right)}^{2} + \textcolor{red}{1} {.5}^{0} {\left(- 3 y\right)}^{3}$

$= 125 - 225 y + 135 {y}^{2} - 27 {y}^{3}$