Binomial theorem
- 2. Expanding Binomials - in this section we will look at ways to expand binomial expressions like: (x + y)5 (2x – 3y)7 We will do this WITHOUT having to multiply the expressions out
- 3. pascal's TrianglE Consider (x + y)n (x + y)0 = 1 which we can think of as 1x0 y0 (x + y)1 = x + y OR 1x1 y0 + 1x0 y1 (x + y)2 = x2 + 2xy + y2 = 1x2 y0 + 2x1 y1 + 1x0 y2 (x + y)3 = x3 + 3x2 y + 3xy2 + y3 OR 1x3 y0 + 3x2 y1 + 3x1 y2 + 1x0 y3 Notice that for each term in the expansion… The exponents add up to “n” Also, for each subsequent term, the exponent for x decreases by 1 while the exponent for y INCREASES by one Additionally, the coefficients for the terms form a pattern known as Pascals’ Triangle (see board)
- 4. To pErform a Binomial Expansion: Go to the appropriate row in Pascal’s triangle to obtain the coefficients Write out terms with the variables, remembering that the powers add up to n for each term, start with xn y0 , end with x0 yn See the example on the next slide
- 5. paTTErns in ThE Expansion of (x + y)n There are n + 1 terms The exponent n of (x+y)n is the exponent of x in the 1st term and the exponent of y in the last term In successive terms, the exponent of x decreases by 1 and the exponent of y increases by 1 The sum of the exponents in each term is n The coefficients are symmetric: They increase at the beginning of the expansion and decrease at the end
- 6. Expand Write row 5 of Pascal’s triangle. 1 5 10 10 5 1 Use the patterns of a binomial expansion and the coefficients to write the expansion of Answer:
- 8. The Binomial Theorem Another way to show the coefficients in a binomial expansion If n is a nonnegative integer, then (a + b)n = 1an b0 + (n/1)an-1 b1 + (n(n-1)/(1*2)an-2 b2 + (n(n-1)(n-2))/(1*2*3) an-3 b3 + … 1a0 bn
- 9. The expression will have nine terms. Use the sequence to find the coefficients for the first five terms. Use symmetry to find the remaining coefficients. Expand
- 10. Answer:
- 11. Expand Answer:
- 12. Factorials The factors in the coefficients of a binomial expansion involve special products called FACTORIALS For example, the product 4 * 3 * 2 * 1 is written 4! and is read “4 factorial” In general, if n is a positive integer, then n! equals n * (n – 1) * (n – 2) * (n – 3) *…2*1 (By definition, 0! = 1) If a rational expression contains some factorials, often a number of terms will cancel out
- 14. Binomial Theorem, facTorial form 0 1 1 2 2 0! ! ! ! ( ) ... !0! ( 1)!1! ( 2)!2! 0! ! n n n n nn n n n x y x y x y x y x y n n n n − − + = + + + + − − 0 ! ( ) * ( )! ! n n n k k k n x y x y n k k − = + = − ∑
- 16. Simplify.
- 17. Answer:
- 18. Expand Answer:
- 19. Finding a speciFic term Sometimes you are only asked to find one term in an expansion Note that when the Binomial expansion is written using Sigma notation, k = 0 for the 1st term, k = 1 for the 2nd term, k = 2 for the 3rd term, and so on. In general, the value of k is one less than the number of the term you are finding!
- 20. Find the fourth term in the expansion of First, use the Binomial Theorem to write the expression in sigma notation. In the fourth term,