Question

Use the Binomial Theorem to expand the binomial and express the result in simplified form.

Answer 1

The Solution:

Given:

`(5x-y)^5`

We are required to use the Binomial Expansion to expand the given expression.

The formula for Binomial expansion is:

`\left(a+b\right)^(n)=\sum_(i=0)^(n)\binom{n}{i}a^(\left(n-i\right))b^(i)`
`\left(5x+(-y)\right)^5=\sum_(i=0)^5\binom{5}{0}(5x)^(\left(5-0\right))(-y)^0`
`\begin{gathered} =(5!)/(0!\left(5-0\right)!)\left(5x\right)^5\left(-y\right)^0+(5!)/(1!\left(5-1\right)!)\left(5x\right)^4\left(-y\right)^1+(5!)/(2!\left(5-2\right)!)\left(5x\right)^3\left(-y\right)^2+ \\ \\ (5!)/(3!\left(5-3\right)!)\left(5x\right)^2\left(-y\right)^3+(5!)/(4!\left(5-4\right)!)\left(5x\right)^1\left(-y\right)^4+(5!)/(5!\left(5-5\right)!)\left(5x\right)^0\left(-y\right)^5 \end{gathered}`
`(5x+(-y))^5=3125x^5-3125x^4y+1250x^3y^2-250x^2y^3+25xy^4-y^5`

Therefore, the correct answer is:

`3125x^5-3125x^4y+1250x^3y^2-250x^2y^3+25xy^4-y^5`