binomial expanding

**Samurai Karasu** · May 13th 2008, 08:25 PM

I'm supposed to use the Binomial Theorem to simplify the two following problems

(X-2)^4 (Answer: X^4-8x^3+24X^2-32X+16

and...

(2+3s)^6 (Answer: 64+576S+2160S^2+4320S^3+4860S^4+2916S^5+729S^6)

But I'm completely stumped, I have absolutely no idea how to use the binomial theorem and am in need of some step by step help. I'd appreciate it tremendously.

**TheEmptySet** · May 13th 2008, 08:41 PM

Originally Posted by Samurai Karasu

I'm supposed to use the Binomial Theorem to simplify the two following problems

(X-2)^4 (Answer: X^4-8x^3+24X^2-32X+16

The binomial theorem states

$(a+b)^n=\sum_{k=0}^{n}\binom{n}{k}(a)^{n-k}(b)^k$

so we we identify the parts a=x b=(-2) and n=4

$(x-2)^4=\sum_{k=0}^{4}\binom{4}{k}(x)^{4-k}(-2)^k$

Now we let k=0,1,2,3,4

$\sum_{k=0}^{4}\binom{4}{k}(x)^{4-k}(-2)^k=\binom{4}{0}(x)^{4-0}(-2)^0+\binom{4}{1}(x)^{4-1}(-2)^1$
$+\binom{4}{2}(x)^{4-2}(-2)^2+\binom{4}{3}(x)^{4-3}(-2)^3+\binom{4}{4}(x)^{4-4}(-2)^4=$

$x^4-8x^3+24x^2-32x+16$

See what you can do with the next one.

**Samurai Karasu** · May 13th 2008, 09:12 PM

I seriously just got it!

The only thing I had to think about was you need to use Pascal's Triangle to put coefficents in front of the numbers, but otherwise I finally got it! Thank you tremendously my man, I think I might be able to do good on that test tomorrow thanks to you.

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