Question

I need help with binomial expansions using Pascal's Triangle. The binomial I am using is (b - 2)^4.

First, I know you set up the problem like this:
1 4 6 4 1

But I don't know where to go from there. Could anyone help me?

Answer 1

Alright so you know you have the co-efficients
\[\{1\phantom{s}4\phantom{s}6\phantom{s}4\phantom{s}1\}\]
So you set up your equation like this:
\[(b-2)^4=(1)(b)^4(-2)^0+(4)(b)^3(-2)^1+(6)(b)^2(-2)^2+4(b)^1(-2)^3+1(b)^0(-2)^4\]

Answer 2

And you and you can simplify so you would have:
\[\eqalign{
(b-2)^4&=(1)(b)^4(-2)^0+(4)(b)^3(-2)^1+(6)(b)^2(-2)^2+4(b)^1(-2)^3+1(b)^0(-2)^4 \\
&=b^4-8b^3+24b^2-32b+16
}\]

Answer 3

Thank you, that helped so much! I'm still kinda confused on the exponents part though, do you just put them in reverse order?

Answer 4

Yeah! Well the thing is if you notice, the exponents of the variable (in this case, \(b\) ) start at the original exponent \((4)\) and decrease to zero. And the exponents of the constant (in this case, the \((-2)\) ) start at zero and work their way up to the original exponent \((4)\). And each term uses their respective Pascalian Binomial Co-efficient one by one!

Answer 5

Thank you so much, I understand it now! :)

Answer 6

Great! Im real happy :)