# Thread: Expanding a cubic expression

1. ## Expanding a cubic expression

I dont know if that's the correct terminology, but I can't find a basic tutorial on solving something like this:

http://i43.tinypic.com/13zux38.jpg

How would you expand that, and can you explain how you knew/what model you used?

Thanks

2. What does that say?

3. hmm you should be able to click and it zooms in

i didnt know how to do the symbols but it says

-5(x + delta x)^3

4. $\displaystyle \left( {x + \Delta x} \right)^3 = x^3 + 3x^2 \Delta x + 3x\left( {\Delta x} \right)^2 + \left( {\Delta x} \right)^3$

5. Ah okay, $\displaystyle -5(x+\Delta x)^{3}$. Ignore the -5 at the front for now. Expanding $\displaystyle (x+\Delta x)^{3}$, you would get $\displaystyle x^{3}+3x^{2}\Delta x+3x(\Delta x)^{2}+(\Delta x)^{3}$.

Notice the descending powers on the first term $\displaystyle x$ and the ascending powers of the second term $\displaystyle \Delta x$. The coefficients of each term (i.e. 1, 3, 3, 1) come from the 4th line of Pascal's triangle, if you know what that is.

Or, the coefficient of the first term in an expansion like this is always $\displaystyle \left( \begin{array}{c} n \\ C \\ 0 \end{array} \right)$
$\displaystyle n$ is the power of the bracket, so in your case, $\displaystyle n=3$. The second coefficient is $\displaystyle \left( \begin{array}{c} n \\ C \\ 1 \end{array} \right)$. This goes on up until $\displaystyle \left( \begin{array}{c} n \\ C \\ n \end{array} \right)$.

The first one comes from $\displaystyle \left( \begin{array}{c} 3 \\ C \\ 0 \end{array} \right)$
Second one from $\displaystyle \left( \begin{array}{c} 3 \\ C \\ 1 \end{array} \right)$
Third one from $\displaystyle \left( \begin{array}{c} 3 \\ C \\ 2 \end{array} \right)$
And fourth one from $\displaystyle \left( \begin{array}{c} 3 \\ C \\ 3 \end{array} \right)$

Then finally, you just multiply each of the terms by that $\displaystyle -5$ we left out to finally get $\displaystyle -5x^{3}-15x^{2}(\Delta x)-15x(\Delta x)^{2}-5(\Delta x)^{3}$

Hope that helps and it's not too much of a wordy explanation. Feel free to reply with any more Qs

6. Thanks for your help, but I'm still confused as to how you even knew to put the $\displaystyle 3x^{2}\Delta x+3x(\Delta x)^{2}$...i was wondering there's a standard form I can go by?

7. This is always true $\displaystyle \left( {a + b} \right)^3 = a^3 + 3a^2 b + 3ab^2 + b^3$.

8. Originally Posted by Plato
This is always true $\displaystyle \left( {a + b} \right)^3 = a^3 + 3a^2 b + 3ab^2 + b^3$.

ahh thank you...that's what i was looking for!

thanks to all who've helped me