The binomial theorem
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The Binomial Theorem. Binomial – two terms. Expand (a + b) 2 (a + b) 3 (a + b) 4. Study each answer. Is there a pattern that we can use to simplify our expressions?. Notice that each entry in the triangle corresponds to a value n C r 0 C 0 1 C 0 1 C 1 2 C 0 2 C 1 2 C 2

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The Binomial Theorem

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The binomial theorem

The Binomial Theorem


The binomial theorem

Binomial – two terms

Expand

(a + b)2

(a + b)3

(a + b)4

Study each answer. Is there a pattern that we can use to simplify our expressions?


The binomial theorem

Notice that each entry in the triangle corresponds to a value nCr

0C0

1C0 1C1

2C02C12C2

3Co3C13C23C3

so we can see that tn,r = nCr = n!/(r!(n-r)!)


The binomial theorem

by Pascal’s formula we can see that

nCr = n-1Cr-1 + n-1Cr

Rewrite the following using Pascal’s Formula

10C4 18C8 + 18C9


The binomial theorem

The coefficients of each term in the expansion of (a + b)n correspond to the terms in row n of Pascal’s Triangle. Therefore you can write these coefficients in combinatorial form.

Lets look at (2a + 3b)3

= 8a3 + 36a2b +54ab2 + 27b3

From Dan

Notice that there is one more term than the exponent number!


The binomial theorem

(a + b)n = nC0an + nC1an-1b+ nC2an-2b2 + … + nCran-rbr + … + nCnbn

or

Expand

(a + b)5

Try it with (3x – 2y)4


The binomial theorem

Factoring using the binomial theorem

Rewrite 1 + 10x2 + 40x4 + 80x6 + 80 x8 + 32x10 in the form (a + b)n

Step 1

We know that there are 6 terms so the exponent must be five


The binomial theorem

Step 2

The first term is 1

Therefore, a =

Step 3

The final term is 32x10

Therefore, b =


The binomial theorem

Homework

Pg 293 # 1ace, 3ab, 4bc, 5ac, 8, 9ace,11ad,12a, 16a, 21


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