Remarks. Remarks. In this section we will learn how to compute the coefficients when we expand a binomial raised to a power.

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1 The Binomial i Theorem In this section we will learn how to compute the coefficients when we expand a binomial raised to a power. ( a+ b) n We will learn how to do this using the Binomial Theorem which can be proved using Mathematical Induction. We have the plague to thank for the binomial theorem. In 665, plague was raging in England, and Isaac Newton, a new graduate of the University of Cambridge, was forced to spend most of the next two years in the relative safety of his family's country manor. During this period, he proved and extended the binomial theorem, invented calculus, discovered the law of universal gravitation, and proved that white light is composed of all colors; all before the age of 5. 3

2 Pirates of Penzance I am the very model of a modern Major General, I've information vegetable, animal, and mineral, I know the kings of England, and I quote the fights historical From Marathon to Waterloo, in order categorical; I'm very well acquainted, too, with matters mathematical, I understand equations, both the simple and quadratical, About binomial theorem I'm teeming with a lot o' news, With many cheerful facts about the square of the hypotenuse. Review - Factorial () If n 0 then n! () If n> 0 the n! n n! n! n n n n 3 3 Example 7! 5 Binomial Coefficient Binomial Coefficient For nonnegative integers n and r, with n > r, a binomial coefficient is defined by n n! r ( n r)! r! This formula can be used to find the coefficient of various terms of a binomial that has been expanded. It can save us a lot of time if we need to know what polynomial you get if we raise a binomial to a large power. It can be very time consuming to multiply out a binomial that is raised to the 5th, 0th, 0th power, etc... 6

3 n n! r ( n r)! r! Another name for this expression is the choose function, and the binomial coefficient of n and r is often read as "n choose r Alternative notations include C(n, r ), n C r and r C n 7 Example 5 Compute 3 5 5! 5! 5 3! ( 5 3 )!3!!3! 3!! 3 8 Example Compute!! 0 9 8! ( )!! 8!! 38!

4 5 Practice Compute 5! 5!! ( ) 5! 3!! 53! 3!! k k and k 0 9 Compute 9 9 Compute 0 9! 9 9!9! ( ) 9! 9 0!0! ( ) 9! 0!9! 9! 9!0! Pascal s Triangle

5 Yang Hui In 3th century, Yang Hui (38-98) presented the arithmetic triangle, which was the same as Pascal's Triangle. In China Pascal's triangle is called "Yang Hui's triangle". 3 Pascal s Triangle can be used to find n C r 5 5! 0! ( 5- )! Using Pascal s Triangle ( a b) 0 + a+ b a+ b a+ b a + ab+ b 3 3 a+ b a + 3a b+ 3ab + b 3 3 a+ b a + a b+ 6a b + ab + b 5 5

6 Binomial Theorem For any positive integer n n n n n n n n n n a+ b a + a b+ a b + a b n n n n + ab + b n n 3 3 The top number of the binomial coefficient is always n, which is the exponent on your binomial. The bottom number of the binomial coefficient starts with 0 and goes up each time until you reach n, which is the exponent on our binomial. 6 Binomial Theorem For any positive integer n + n n n n n n n n + ab + b n n + n n n n n 3 3 ( a b) a a b a b a b The st term of the expansion has a (first term of the binomial) raised to the n th power, which is the exponent on our binomial. From there a s exponent goes down, until the last term, where it is being raised to the 0 power; which is why we don t see it written. 7 Binomial Theorem For any positive integer n + n n n n n n n n + ab + b n n + n n n n n 3 3 ( a b) a a b a b a b The first term of the expansion has b (second term of the binomial) raised to the 0 power, which is why we don t see it written. From there b s exponent goes up, until the last term, where it is being raised to the n th power, which is the exponent on our binomial. 8 6

7 Example ( x + 3y) ( 3 ) ( 3 ) 5 3 x x y x y ( 3 ) ( 3 ) ( 3 ) 3 5 x y x y y a x b 3 y n () x ( 5) x ( 3y) ( 0) x ( 9y ) 3 5 ( 0) x ( 7y ) ( 5) x( 8y ) ()( 3y ) x + 5xy+ 90xy + 70xy + 05xy + 3y Example ( x)( y) + ( y) 3 x y a x b y n ( x) ( x) ( y) ( x) ( y) 3 ( )( ) ( ) ( 6)( ) ( ) ( ) ( 6) x ( 8) xy 3 ( 6) xy xy 3 y 3 3 x x y+ x y x y + y + + 6x 3xy+ xy 8xy + y Practice Expand ( x + 3y ) 3 3 a+ b a x b y We have form with, x y x y x y x y x + 3x 3y + 3x 3y + 3y ( 3 ) + ( 3 ) + ( 3 ) + ( 3 ) x + 9xy + 7xy + 7y 3 6 7

8 Finding a Particular Term in a Binomial Expansion The r th term of the expansion of n a r b n r+ r ( a+ b) n is The top number of the binomial coefficient is n, which is the exponent on our binomial. The bottom number of the binomial coefficient is r -, where r is the term number. Finding a Particular Term in a Binomial Expansion The r th term of the expansion of n a r b n r+ r ( a+ b) n is a is the first term of the binomial and its exponent is n - r +, where n is the exponent on the binomial and r is the term number. b is the second term of the binomial and its exponent is r -, where r is the term number. 3 Find the fifth term of the Example 9 expansion of x a x b n 9 r ( x ) ( x ) 5 ( 6) 0 x x 0 8

9 Practice Find the fourth term of the ( x+ y) 7 expansion of a x b y n 7 r 7 7 ( ) x + y 7 3 x y ( 35) x ( 8) y x y 3 5 TI 83/8 users may want to download the free program binexpan.zip from This application will compute all of the coefficients in a binomial expansion up to the 9 th degree This ends our Algebra Lectures You are now a Super-Algebra Student 7 9