10.2b - Binomial Theorem.

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Presentation transcript:

10.2b - Binomial Theorem

Consider the expansion: What are some different patterns you see in the expansion above?

What are some different patterns you see? # of terms in expansion is one more than the binomials power 2. Powers of x go from n0; powers of y from 0n 3. 1st and last coefficients is one 4. Term # is always one more than the power of y

Each row is the coefficients for n = row # Fill in the coefficients of the (above) expansions below: Row:#___ n = 0 1 n = 1 Row:#___ 1 1 1 1 2 1 Row:#___ 2 n = 2 1 3 3 1 Row:#___ 3 n = 3 1 4 6 4 1 Row:#___ 4 n = 4 1 5 10 10 5 1 n = 5 Row:#___ 5 n = 6 1 6 15 20 15 6 1 Row:#___ 6 Each row is the coefficients for n = row #

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 Pascal’s Triangle This is called: ___________ _________ The ‘math stud’ who is given credit for this is: _____________________________ Blaise Pascal (1623 – 1662)

Binomial Theorem The binomial expansion of Where the coefficients of each term are given by: n = power of binomial m = the “y” exponent

1. Expand the binomial: Find the coefficients using Pascal’s triangle Label the same coefficients using the binomial theorem a) b)

2. Find the binomial coefficients of the given terms using the binomial theorem (check your answers with the answers you got in #1) 3 15 a) 3 15 b)

6 2 2 20 c) y3 d) The fourth term of the binomial: 2. Find the binomial coefficients of the given terms using the binomial theorem (check your answers with the answers you got in #1) c) 6 y3 d) The fourth term of the binomial: 2 2 20

2. Find the binomial coefficients of the given terms using the binomial theorem (check your answers with the answers you got in #1) e) 1

3. Expand the binomial: Hint: a) write out: (x + y)5 b) substitute _____ in for y c) simplify each term -2

Coef x y Term (Pascal) 1 x5 (-2)0 = 1 x5 5 x4 (-2)1 = -2 -10x4 10 x3 (-2)2 = 4 40x3 x2 (-2)3 = -8 10 -80x2 x1 (-2)4 = 16 80x1 5 x0 = 1 (-2)5 = -32 -32 1

3 4 84 4. Find the fourth term of the binomial: Coef x y Term (binom) (-1)3 = -1 -84m6 x9 x8y x7y2 x6y3 3 4 84

5. Expand the binomial: Hint: a) write out: (2a + 3b)4 b) substitute _____ for x & _____ for y c) simplify each term 2a 3b

Coef x y Term (Pascal) 1 (2a)4 = 16a4 (3b)0 = 1 16a4 4 (2a)3 = 8a3 (3b)1 = 3b 96a3b 6 (2a)2 = 4a2 (3b)2 = 9b2 216a2b2 4 (2a)1 = 2a (3b)3 = 27b3 216ab3 1 (2a)0 = 1 (3b)4 = 81b4 81b4