Question

I need help with the calculus portion of my ACT prep guide

Answer 1

The first thing we have to do is find the pascal coefficients of the triangle for this we use the following image

From our exercise we know that

$\begin{gathered} (2x^3-3y)^3\to(a+b)^3 \\ a=2x^3 \\ b=-3y \end{gathered}$

Then our coefficients of Pascal's triangle are 1 - 3 - 3 - 1

Using the right triangle we get:

$1a^3+3a^2b+3ab^2+1b^3$

We substitute the values of a and b in our new expression to find our solution

$1(2x^3)^3+3(2x^3)^2(-3y)+3(2x^3)(-3y)^2+1(-3y)^3$$\begin{gathered} 8x^9+3(4x^6)^{}(-3y)+3(2x^3)(9y^2)^{}-27y^3 \\ 8x^9-36x^6y+54x^3y^2-27y^3 \end{gathered}$

So the solution is:

$8x^9-36x^6y+54x^3y^2-27y^3$