11.6 Binomial Theorem & Binomial Expansion

Pascal’s Triangle Pascal’s Triangle Patterns – (1) Sums of Rows: sum of elements in row n is 2^n; (2) Prime Number – If 1st element in a row is prime (0th element is “1”), then all elements in that row are divisible by it; (3) Hockey Stick pattern; (4) Magic 11’s; (5) Fibonacci’s Sequence. Check out http://ptri1.tripod.com/.

Pascal’s Triangle with even and odd numbers colored differently:

Binomial Expansion Patterns: (1) # of terms is 1 greater than the power; (2) exponent on “a” decreases while exponent on “b” increases; (3) sum of exponents on any terms equals the power; (4) exponent on “b” is always 1 less than the number of the term (ex.: 4th term in expansion of (a+b)^5 has exponent of 3 on “b”); and (5) coefficients come from corresponding row of Pascal’s Triangle.

Pascal's Triangle Use this triangle to expand binomials of the form (a+b)n. Each row corresponds to a whole number n. The first row consists of the coefficients of (a+b)n when n = 0. Example 1 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 1 7 21 35 35 21 7 1 1 8 28 56 70 56 28 8 1 1 9 36 84 126 126 84 36 9 1 Example 2

Expand (a + b)6 1 6 15 20 15 6 1 (coefficients from Pascal’s triangle) 1a6 6a5 15a4 20a3 15a2 6a1 1a0 (exponents of a begin with 6 and decrease) 1a6b0 6a5b1 15a4b2 20a3b3 15a2b4 6a1b5 1a0b6 (exponents of b begin with 0 and increase by 1) a6 + 6a5b + 15a4b2 + 20a3b3 + 15a2b4 + 6ab5 + b6 (expansion in standard, simplified form) Return to Triangle

Expand (x – 2)3 (let a = x and b = –2) 1 3 3 1 (coefficients from Pascal’s triangle) 1x3 3x2 3x1 1x0 1x3(–2)0 3x2(–2)1 3x1(–2)2 1x0(–2)3 x3 – 6x2 + 12x – 8 (expansion in standard, simplified form) (do example 2 in text, p. 850) Return to Triangle

Factorial Notation Read as “n factorial”

Finding a particular term in a Binomial Expansion: Formula for finding a particular term in expansion of (a + b)n is: Ex.: Find the 4th term in expansion of (a + b)9: This is the 4th term, so value of r (b’s exponent) is 4 – 1 = 3. This means the exponent for a is 9 – 3, or 6. So, we have the variables of the 4th term: a6b3 Coefficient is:

Finding a particular term in a Binomial Expansion: Formula for finding a particular term in expansion of (a + b)n is: Ex.: Find the 8th term in expansion of (2x – y)12: This is 8th term, exponent for b is 8 – 1 = 7. This means the exponent for a is 12 – 7, or 5. So, the variables of the 8th term: a5b7, or (2x)5(–y)7. Coefficient:

Binomial Theorem: (do example 4 in text, p. 853)