Question

Hello,Can you help me with Number 13? Is there a shorter way of doing these? Use the binomial and express the results in simplified form

Answer 1

Case: Binomial expansion theorem.

Method:

`(a+b)^n=^nC_0a^nb^0+^nC_1a^(n-1)b^1+^nC_2a^(n-2)b^2+...+^nC_na^0b^n`

The given question is:

`\begin{gathered} (5x-1)^3 \\ a=5x,b=-1 \\ (5x-1)^3 \\ ^3C_0(5x)^3(-1)^0+^3C_1(5x)^2(-1)^1+^3C_2(5x)^1(-1)^2+^3C_3(5x)^0(-1)^3 \end{gathered}`

Evaluating this:

`\begin{gathered} 1*(125x^3)(1)+3(25x^2)(-1)+3(5x)(1)+1(1)(-1) \\ 125x^3-75x^2+15x-1 \end{gathered}`

Final answer:

`125x^(3)-75x^(2)+15x-1`