Helpful Tools for Calculus, Chapter 2: Helpful Tools for the Mathematics of Calculus
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Pascal’s Triangle
A common type of problem in mathematics is that of expanding a binomial 〖(a+b)〗^n. Beginning algebra students are shown how to distribute, FOIL and expand polynomials through multiplication, but as we get further into mathematics some of the problems get more challenging.
One helpful trick is to use Pascals Triangle to expand a binomial. Let us first construct Pascals Triangle using the following technique. Begin with a 1 in the center, this is row 0. The next row put two 1’s diagonally away from the first 1 in a triangular shape, this is row 1. For the next row and all of the following rows, begin with a 1 diagonally below and to the left of the previous row. Each additional term is defined by adding the two terms above until you get to the end of the row which ends with 1. Every row should add one new term and always begin and end with a 1.
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
1 6 15 20 15 6 1
1 n … n 1
Binomial Expansion
But, how does this help with algebra? The answer is that these numbers represent the coefficients when expanding a binomial. For 〖(a+b)〗^n we take n as the row number. Then each number in that row represents to coefficient for a different term. To construct the terms begin by taking a^n b^0 then for each term decrease the exponent of a by 1 and increase the exponent of b by 1. By the last term we should have a^0 b^n. Recall that anything, except 0, raised to the 0th power is equal 1, so you can simplify.
Example 1: Expand the following binomial:
〖( x+y )〗^5
Solution:
The exponent is 5 so we should refer to the row whose second number is 5: 1 5 10 10 5 1.
These form the coefficients of the expansion:
1x^5 y^0+5x^4 y^1+10x^3 y^2+10x^2 y^3+5x^1 y^4+1x^0 y^5
We often omit the variables whose exponent is 0:
x^5+5x^4 y+10x^3 y^2+10x^2 y^3+5xy^4+y^5
Example 2: Expand the following binomial:
〖( 2a-b )〗^4
Solution:
The exponent is 4 so we should refer to the row whose second number is 5: 1 4 6 4 1.
These form the coefficients of the expansion:
x^4+4x^3 y+6x^2 y^2+4xy^3+y^4
Then we substitute the values of x and y:
〖(2a)〗^4+4(2a)^3 (-b)+6〖(2a)〗^2 〖(-b)〗^2+4(2a)〖(-b)〗^3+〖(-b)〗^4
Simplify:
〖16a〗^4-4∙8a^3 b+6∙4a^2 b^2-4∙2ab^3+b^4
〖16a〗^4-32a^3 b+24a^2 b^2-8ab^3+b^4