SFM Productions Presents: Another adventure in your Pre-Calculus experience! 9.5The Binomial Theorem

Homework for section 9.5 P686#7-11, 27-33, 39 (use Pascal’s triangle, even though it says to use the Binomial Theorem), 47-51, 55-59, 85, 87

Binomials are polynomials that have two terms. We will study a formula that gives a quick method of raising a binomial to a power.

1 What type of patterns or observations can be made?

In each expansion, there are n+1 terms. In each expansion, x and y have symmetrical roles. The powers of x decrease by 1 in successive terms, while the powers of y increase by 1 in successive terms. The sum of the powers of each term is n. The first term is always x n ; the last term is always y n The coefficients increase and then decrease in a symmetrical pattern.

The hard part is trying to figure out all the coefficients. That’s where The Binomial Theorem is helpful …… In the expansion of Where n C r is the coefficient n- r is the exponent of the x term, and r is the exponent of the y term.

It’s also a thingi on your calculator … Sometimes,is written like this: They both mean the same thing.

Note how n C r is the same as n C n- r

Another way to find coefficients is by using: Pascal’s Triangle 1 x + y x 2 +2xy + y 2 x 3 + 3x 2 y + 3xy 2 +y 3 x 4 +4x 3 y +6x 2 y 2 +4xy 3 +y 4 x 6 +6x 5 y+15x 4 y 2 +20x 3 y 3 +15x 2 y 4 +6xy 5 +y 6 x 5 +5x 4 y +10x 3 y 2 +10x 2 y 3 +5xy 4 +y 5

Pascal’s Triangle

The term for writing out the coefficients of a binomial Raised to a power is: expanding a binomial or binomial expansion. Ex: Write the expansion for: Pascal’stells us that the coefficients are: 1, 4, 6, 4, x4x4 + x3x3 (1) 1 + x2x2 (1) 2 (1) 3 (1) 4 x + + Alternate signs : start with positive first term, then alternate neg/pos..

Write the expansion for: 4641 (2x) 4 + (2x) 3 (3) 1 + (2x) 2 (3) 2 (3) 3 (3) 4 (2x) + + 1

Finding a term in a Binomial Expansion Ex: Find the 6 th term of the binomial expansion of: For 1 st term: n=8, r=0 8 C 0 (a) 8-0 (2b) 0 For 2 nd term: n=8, r=1 8 C 1 (a) 8-1 (2b) 1 Therefore …… For 6 th term: n=8, r=5 8 C 5 (a) 8-5 (2b) 5 56 a b a 3 b 5

Find the 9 th term of the binomial expansion of: 9 th term means that r = ? r = 8 n = ? n = C 8 (3a) 12-8 (2b) a 4 b 8 In this example, what is x? In this example, what is y? x = 3a y = 2b (x + y) 12 (495)(3 4 )(a 4 )(2 8 )(b 8 ) 12 C 8 (x) 12-8 (y) 8

Go! Do!