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

Solution:

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:

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

Final answer:

$125x^{3}-75x^{2}+15x-1$