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Chapter 6: Polynomials

6.9 Pascal’s Triangle and Binomial Expansion

Pascal’s triangle (1653) has been found in the works of mathematicians dating back before the 2nd century BC. While Pascal’s triangle is useful in many different mathematical settings, it will be applied to the expansion of binomials. In this application, Pascal’s triangle will generate the leading coefficient of each term of a binomial expansion in the form of:

(a+b)n

Yang Hui's triangle

For example:

(a+b)2=a2+2ab+b2(1+2+1)(a+b)3=a3+3a2b+b3(1+3+3+1)

Pascal’s Triangle

(a+b)01201(ab)0(a+b)11+12111(ab)1(a+b)21+2+12212+1(ab)2(a+b)31+3+3+12313+31(ab)3(a+b)41+4+6+4+12414+64+1(ab)4(a+b)51+5+10+10+5+12515+1010+51(ab)5(a+b)61+6+15+20+15+6+12616+1520+156+1(ab)6(a+b)71+7+21+35+35+21+7+12717+2135+3521+71(ab)7

The generation of each row of Pascal’s triangle is done by adding the two numbers above it.

1Start with 11+1The outside number is always 11+2+1The two 1’s in the last row add to 21+3+3+11+2 above adds to 31+4+6+4+11+5+10+10+5+11+6+15+20+15+6+11+7+21+35+35+21+7+1We can extend Pascal’s triangle using this1+8+28+56+70+56+28+8+1(a+b)81+9+36+84+126+126+84+36+9+1(a+b)9

Example 6.9.1

Use Pascal’s triangle to expand (a+b)9.

The variables will follow a pattern of rising and falling powers:

a9+a8b+a7b2+a6b3+a5b4+a4b5+a3b6+a2b7+ab8+b9

When we insert the coefficients found from Pascal’s triangle, we create:

a9+9a8b+36a7b2+84a6b3+126a5b4+126a4b5+84a3b6+36a2b7+8ab8+b9

Problem: Use Pascal’s triangle to expand the binomial (a+b)12.

A Visual Representation of Binomial Expansion

(a+b)1=a+b. (a+b)2 = a2 + 2ab+ b2 (a+b)2=a2+3a+2b+dab2+b2, (a+b)4=a4+4a2b+6a2b2+4ab2+b2

The fourth expansion of the binomial is generally held to represent time, with the first three expansions being width, length, and height. While we live in a four-dimensional universe (string theory suggests ten dimensions), efforts to represent the fourth dimension of time are challenging. Carl Sagan describes the fourth dimension using an analogy created by Edwin Abbot (Abbot: Flatland: A Romance of Many Dimensions). A video clip of Sagan’s “Tesseract, 4th Dimension Made Easy”  can be found on YouTube.

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