How do you use the Binomial Theorem to expand #(x + 1)^4#?

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May 31, 2016

Answer:

#(x+1)^4 = x^4 + 4x^3+6x^2+4x+1#

Explanation:

To use the binominal theorem we must know what Pascals triangle is because it gives us the binominal coefficients.

Wikipedia

Note that the top "1" is considered row no 0 by convention.
So to get our coefficients we go to the 4th row and get our seqence of #1,4,6,4,1#.

that means that we will have to multiply each x with the corresponding value from the sequence.

In our answer we can see this:

#(x+1)^4 = x^4 + 4x^3+6x^2+4x+1#

All the numbers are the same as our sequence (but the "1's" have been removed in the answer since #x^4*1 = x^4# and #1*1 = 1#

the next thing to do is to raise our #y-value# to the power of #n/0, n/1, etc # until we get to #n/n# but since #1^n = 1# we don't need to do anything except for the last row where we have our #+1#.

It all becomes much more evident if we exchange the #1# for #y#
giving us:

#(x+y)^4= x^4 + 4 x^3y + 6 x^2 y^2 + 4 x y^3 + y^4.#

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