Question

Expand using binomial theorem(3a-2)^3

Answer 1

Recall that the binomial theorem states that:

`(a+b)^n=\sum ^n_(k=0){\binom{n}{k}}a^(n-k)b^k\text{.}`

Therefore:

`\mleft(3a-2\mright)^3=\sum ^3_(k=0){\binom{3}{k}}(3a)^(3-k)(-2)^k.`

Then:

`\begin{gathered} (3a-2)^3={\binom{3}{0}}(3a)^(3-0)(-2)^0+{\binom{3}{1}}(3a)^(3-1)(-2)^1+{\binom{3}{2}}(3a)^(3-2)(-2)^2 \\ +{\binom{3}{3}}(3a)^(3-3)(-2)^3\text{.} \end{gathered}`

Simplifying the above result we get

`\begin{gathered} (3a-2)^3=(3a)^3+3(3a)^2(-2)+3(3a)(-2)^2+(-2)^3 \\ =27a^3-54a^2+36a-8. \end{gathered}`