Question

could some one give me an alternative for the binomial expansion theorem?

Answer 1

Alternative? Are you expanding a binomial?

Answer 2

Well, I'm making up a class and I need to solve this:
"In your lab, a substance's temperature has been observed to follow the function T(x) = (x - 2)3 + 8. The turning point of the graph is where the substance changes from a solid to a liquid. Explain to your fellow scientists how to find the turning point of this function, using complete sentences."
pretty much everywhere it says I should use the Binomial Expansion Theorem but we haven't gone over that, yet. I'm making up Algebra 2. I don't know what else to do?

Answer 3

The theorem gives you the general case for expanding a binomial, so that you can write \((x+y)^n\) as a sum of terms.

The theorem itself is
\[(x+y)^n=\sum_{k=0}^n\binom nk x^ky^{n-k}\]
Since \(n=3\) is manageable and small, you can just multiply:
\[(x-2)^3=(x-2)(x-2)(x-2)=x^3-6x^2+12x-8\]

Answer 4

In terms of the theorem, you would say
\[\begin{align*}(x-2)^3&=\binom30x^0(-2)^{3-0}+\binom31x^1(-2)^{3-1}+\binom32x^2(-2)^{3-2}+\binom33x^3(-2)^{3-3}\\
&=1x^0(-2)^{3}+3x(-2)^{2}+3x^2(-2)^{1}+1x^3(-2)^{0}\\
&=-8+12x-6x^2+x^3\\

\end{align*}\]

Answer 5

If you weren't familiar with the notation,
\[\binom nk={}_{n}C_{k}=\frac{n!}{k!(n-k)!}\]

Answer 6

Thank you dude!!!

Answer 7

yw