Chapter 8 Section 8.0 Binomial Expansion and Pascal's … 8... · 8.0 Notes.notebook August 16, 2016 Consider some results of expanding the binomial expression : etc. Note the following

8.0 Notes.notebook August 16, 2016

Learning Objectives:1) To expand binomials of the form (ax + by)n 2) To reproduce Pascal's Triangle3) Apply Pascal's Triangle to binomial expansion

Chapter 8

Section 8.0 Binomial Expansion

and Pascal's Triangle


Binomial Expansion

(x+y)0 =(x+y)1 =

(x+y)2 =

(x+y)3 =

(x+y)4 =

Polynomial: A single term or the sum of two or more terms containing variables with wholenumber exponents.

Binomial: A polynomial with two terms.

What patterns in the answers do you notice?


It is named after French mathematician Blaise Pascal (1600's), but the same array of numbers actually appears in a Chinese document printed 300 years before Pascal.

Pascal's Triangle is a mathematical triangular array.

In addition to being useful in binomial expansions, the triangle has many uses in probability.

Pascal's Triangle is also important because of the many different interesting and important number patterns which have been shown to be present in the triangle.


Row 0

Row 1Row 2Row 3

Row 4

Row 7

Row 6Row 5

Fill in the triangle

Obj. 1: Reproduce Pascal's Triangle

The Right and Left sides of the triangle consist entirely of 1's.Each interior entry in the triangle is the sum of the two entries above.


Row 0

Row 1Row 2Row 3

Row 4

Row 7

Row 6Row 5

Fill in the triangle

Obj. 1: Reproduce Pascal's Triangle


Notice anything?


Binomial Expansion

(x+y)0 =(x+y)1 =

(x+y)2 =

(x+y)3 =

(x+y)4 =




1x + yx2 + 2xy + y2

x3 + 3x2y + 3xy2 + y3

x4 + 4x3y + 6x2y2 + 4xy3 + y4


Binomial Expansion

(x+y)0 =(x+y)1 =

(x+y)2 =

(x+y)3 =

(x+y)4 =




11x + 1y1x2 + 2xy + 1y2

1x3 + 3x2y + 3xy2 + 1y3

1x4 + 4x3y + 6x2y2 + 4xy3 + 1y4


In Pascal's Triangle, each row represents the corresponding binomial coefficients of the binomial series.



Consider some results of expanding the binomial expression :

etc.


etc.

Note the following patterns for the variable exponents:

1. The first term when expanding (a + b)n is an .The exponents on a decrease by 1 in each successive term.

2. The exponents on b increase by 1 in each successive term. The first term does not have b (since the exponent is 0 and b0 = 1). The last term is bn.

3. The sum of the exponents on the variables on every term

in the expansion of (a + b)n is equal to n.

4. The number of terms in the expansion is one greater than the power of the binomial, n. There are n + 1 terms for an expansion of (a + b)n .

Also note :




etc.

Note the following patterns for the variable exponents:

1. The first term when expanding (a + b)n is an .The exponents on a decrease by 1 in each successive term.

2. The exponents on b increase by 1 in each successive term. The first term does not have b (since the exponent is 0 and b0 = 1). The last term is bn.

3. The sum of the exponents on the variables on every term

in the expansion of (a + b)n is equal to n.

4. The number of terms in the expansion is one greater than the power of the binomial, n. There are n + 1 terms for an expansion of (a + b)n .

Also note :


Using Pascal's triangle and the pattern of exponents, expand

Obj. 2: Apply Pascal's Triangle to Binomial Expansion

3a) (2x + 3)4



Using Pascal's triangle and the pattern of exponents, expand

3b) (2x3 3y)5

Obj. 2: Apply Pascal's Triangle to Binomial Expansion



Homework: Worksheet

Documents

Chapter 8 Section 8.0 Binomial Expansion and Pascal's … 8... · 8.0 Notes.notebook August 16, 2016 Consider some results of expanding the binomial expression : etc. Note the following