http://www.ck12.org Chapter 2. Polynomials and Rational Functions
Example A
Expand the following binomial using Pascal’s Triangle:( 3 x− 2 )^4
Solution: The coefficients will be 1, 4, 6, 4, 1; however, since there are already coefficients with thexand the
constant term you must be particularly careful.
1 ·( 3 x)^4 + 4 ·( 3 x)^3 ·(− 2 )+ 6 ·( 3 x)^2 ·(− 2 )^2 + 4 ·( 3 x)·(− 2 )^3 + 1 ·(− 2 )^4
Then it is only a matter of multiplying out and keeping track of negative signs.
81 x^4 − 216 x^3 + 216 x^2 − 96 x+ 16
Example B
Expand the following trinomial:(x+y+z)^4
Solution: Unfortunately, Pascal’s triangle does not apply to trinomials. Instead of thinking of a two dimensional
triangle, you would need to calculate a three dimensional pyramid which is called Pascal’s Pyramid. The sum of
all the terms below is your answer.
1 x^4 + 4 x^3 z+ 6 x^2 z^2 + 4 xz^3 + 1 z^4
4 x^3 y+ 12 x^2 yz+ 12 xyz^2 + 4 yz^3
6 x^2 y^2 + 12 xy^2 z+ 6 y^2 z^2
4 xy^3 + 4 y^3 z
1 y^4Notice how many patterns exist in the coefficients of this layer of the pyramid.
Example C
Expand the following binomial:(^12 x− 3 )^5
Solution: You know that the coefficients will be 1, 5, 10, 10, 5, 1.