Multiplying Binomials Methods

[Guest]

Is there a faster/another method to multiply big binomials like (a+b)^6? I guess you would call it "expanding binomials" or something.

I'm really prone to errors when doing the regular manual method. And the binomial theorem and the pascal triangle is just driving me nuts.

 

[galactus]

You can try it this way. I learned this somewhere years back.

Let's say we want to expand \(\displaystyle (a+b)^{6}\)

No Pascal and no Binomial.

OK. The first term, as you know, is a^6.

Now, the next coefficient is the exponent of that 6.

So, we have, so far: \(\displaystyle a^{6}+6a^{5}b\)

The next coefficient is the preceding coefficient times the exponent of 'a' divided by one plus the exponent of b. From 6a^5b, 6 is the coefficient; the exponent of 'a' is 5; the exponent of b is 1, so we add 1 and get 2: (6*5)/2=15

So, we have \(\displaystyle a^{6}+6a^{5}b+15a^{4}b^{2}\)

Next, (15*4)/3=20.

\(\displaystyle a^{6}+6a^{5}b+15a^{4}b^{2}+20a^{3}b^{3}\)

Now, the pattern repeats. (20*3)/4=15

The exponents of a and b are easy to remember. One goes up, the other goes down. This is a slick way to find the coefficients without Pascal or the combinations.

See?.