Question

Expand: (3a - 4b)^8. Write a brief explanation of your process for finding the expansion of this binomial.

Answer 1

To expand the binomial

`(3a-4b)^8`

We can use the binomial theorem.

According to the binomial theorem, our binomial can be rewritten as:

`(3a-4b)^8=\sum_{n\mathop{=}0}^8\begin{pmatrix}8 \\ n\end{pmatrix}(3a)^(8-n)(-4b)^n`

Then, expanding this sum, we have:

`\begin{gathered} (3a-4b)^8=(8!)/(0!(8-0)!)(3a)^8(-4b)^0+(8!)/(1!(8-1)!)(3a)^7(-4b)^1+... \\ +(8!)/(7!(8-7)!)(3a)^1(-4b)^7+(8!)/(8!(8-8)!)(3a)^0(-4b)^8 \\ \\ =6561a^8-69984a^7b-870912a^5b^3+1451520a^4b^4-1548288a^3b^5 \\ +1032192a^2b^6-393216ab^7+65536b^8 \end{gathered}`