Question

In the following exercises, expand each binomial ${\left(y-3\right)}^4$ .

Short answer

The Binomial expansion is $y^4-12y^3+54y^2-108y+81$.

Step-by-step solution

Step 1. Given Information.

The expression is ${\left(y-3\right)}^4$.

Step 2. Apply Binomial Theorem

Binomial Theorem is :-${\left(a+b\right)}^n=\sum _{i=0}^n\left(\genfrac{}{}{0ex}{}{n}{i}\right)a^{\left(n-i\right)}{b}^i$

Step 3. Solve for Binomial Theorem .

Using Binomial Theorem , we get:-

$\sum _{i=0}^4\left(\genfrac{}{}{0ex}{}{4}{i}\right)y^{\left(4-i\right)}{\left(-3\right)}^i$

Expanding :-

$\frac{4!}{0!\left(4-0\right)!}{y}^4{\left(-3\right)}^0+\frac{4!}{1!\left(4-1\right)!}{y}^3{\left(-3\right)}^1+\frac{4!}{2!\left(4-2\right)!}{y}^2{\left(-3\right)}^2+\frac{4!}{3!\left(4-3\right)!}y{}^1{\left(-3\right)}^3+\frac{4!}{4!\left(4-4\right)!}{y}^0{\left(-3\right)}^4$

Simplifying the terms ,we get:-

$y^4-12y^3+54y^2-108y+81$