1. The Binomial Theorem
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2. Expanding Binomials
Expanding a binomial such as (a + b)n means to write the factored form as a sum.
(a + b)0 = 1 (a + b)1 = a + b (a + b)2 = a2 + 2ab + b2 (a + b)3 = a3 + 3a2b + 3ab2 + b3 (a + b)4 = a4 + 4a3b + 6a2b2 + 4ab3 + b4 (a + b)5 = a5 + 5a4b + 10a3b2 + 10a2b3 + 5ab4 + b5
1 term
2 terms
3 terms
4 terms
5 terms
6 terms
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3. Expanding Binomials
(a + b)0 = 1 (a + b)1 = a + b (a + b)2 = a2 + 2ab + b2 (a + b)3 = a3 + 3a2b + 3ab2 + b3 (a + b)4 = a4 + 4a3b + 6a2b2 + 4ab3 + b4 (a + b)5 = a5 + 5a4b + 10a3b2 + 10a2b3 + 5ab4 + b5
The expansion of (a + b)n contains n + 1 terms.
The first term is an and the last term is bn.
The powers of a decrease by 1 for each term; the powers of b increase by 1 for each term.
The sum of the exponents of a and b is n.
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4. Pascal’s Triangle
There are also patterns in the coefficients of the terms. When written in a triangular array, the coefficients are called Pascal’s triangle.
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5. Pascal’s Triangle (a + b)0 (a + b)1 (a + b)2 (a + b)3 (a + b)4
n = 0
n = 1
n = 2
n = 3
n = 4
n = 5 1 5 6 1 6 15 20 15 6 1
Add the consecutive numbers in the row for n = 5 and write each sum “between and below” the pair.
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6. Pascal’s Triangle Example: Expand (a + b)7.
Use n = 7 row of Pascal’s triangle as the coefficients and the noted patterns.
n = 6
n = 7
(a + b)7 = 1a7 + 7a6b + 21a5b2 + 35a4b3 + 35a3b4
+ 21a2b5 + 7ab6 + 1b7
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7. Factorials Factorial of n: n! If n is a natural number, then
An alternative method for determining the coefficients of (a + b)n is based on using factorials.
The factorial of n, written n! (read “n factorial”), is the product of the first n consecutive natural numbers.
Factorial of n: n!
If n is a natural number, then
n! = n(n – 1)(n – 2)(n – 3) ∙ 3 ∙ 2 ∙ 1.
The factorial of 0, written 0!, is defined to be 1.
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8. Evaluating Factorials
Example:
Evaluate each expression.
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9. Binomial Theorem Binomial Theorem If n is a positive integer, then
It can be proved that the coefficients of terms in the expansion of (a + b)n can be expressed in terms of factorials. Following the earlier patterns and using the factorial expressions of the coefficients, we have the binomial theorem.
Binomial Theorem
If n is a positive integer, then
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10. Binomial Theorem Example: Use the binomial theorem to expand (x + 3)4.
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11. Binomial Theorem Example:
Use the binomial theorem to expand (3a – 5b)6.
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12. Binomial Expansion (r + 1)st Term in a Binomial Expansion
The (r + 1)st term of the binomial expansion of (a + b)n is
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13. Binomial Expansion Example:
Find the ninth term in the expansion of (3x – 5y)10.
n = 10, a = 3x, b = – 5y, r + 1 = 9, therefore r = 8
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