X represent a number or letter a represent a number or letter

The Binomial

Binomials

x represent a number or letter

a represent a number or letter

n represents a power

For example: (1 + x)2 = or (x + 2)3 =

Questions… Find Multiply

Expand the binomial What is..?

Binomial Theorem

part 1: (Pascal ∆)

Fill in the missing numbers for this triangle on your paper

Describe the patterns

part 1: (Pascal ∆)

20 = 1 21 = 1+1 = 2 22 = 1+2+1 = 4 23 = 1+3+3+1 = 8 24 = 1+4+6+4+1 = 16

The Sums of the Rows

The sum of the numbers in any row is equal to 2 to the nth power or 2n, when n is the number of the row. For example:  

The sum of the numbers in any row is equal to ___ to the nth power or ____ when n is ___________________.

2

2n

the number of the row

A binomial is a polynomial with two terms such as x + a. Often we need to raise a binomial to a power. In this section we'll explore a way to do just that without lengthy multiplication.

Can you see a pattern?

Can you make a guess what the next one would be?

We can easily see the pattern on the x's and the a's. But what about the coefficients? Make a guess and then as we go we'll see how you did.

Let's list all of the coefficients on the x's and the a's and look for a pattern.

1

1 1

1 2 1

1 3 3 1

1 4 6 4 1

+

+

+

+

+

+

Can you guess the next row?

1 5 10 10 5 1

+

+

+

+

1

1 1

1 2 1

1 3 3 1

1 4 6 4 1

1 5 10 10 5 1

This is called Pascal's Triangle and would give us the coefficients for a binomial expansion of any power if we extended it far enough.

This is good for lower powers but could get very large. We will introduce some notation to help us and generalize the coefficients with a formula based on what was observed here.

Patterns observed Consider the patterns formed by expanding (x + y)n.

• Symmetry of coefficients (i.e. 2nd term and 2nd to last term have same coefficients, 3rd & 3rd to last etc.) so once you've reached the middle, you can copy by symmetry rather than compute coefficients.

• Powers on x and y add up to power on binomial

• x's increase in power as y's decrease in power from term to term.

(x + y)0 = 1

(x + y)1 = x + y

(x + y)2 = x2 + 2xy + y2

(x + y)3 = x3 + 3x2y + 3xy2 + y3

(x + y)4 = x4 + 4x3y + 6x2y2 + 4xy3 + y4

(x + y)5 = x5 + 5x4y + 10x3y2 + 10x2y3 + 5xy4 + y5

1. The exponents on x decrease from n to 0 and the exponents on y increase from 0 to n.

2. Each term is of degree n.

Example: The 4th term of (x + y)5 is a term with x2y3.”

Binomial Coefficients

The coefficients of the binomial expansion are called binomial coefficients. The coefficients have symmetry.

Example 2: What are the last 2 terms of (x + 2)10 ?

Since n = 10, the last two terms are 10x(2)9 + 1y10.

Use a calculator to calculate (2)9 = 510 then multiply it by 10

Your final answer should be 5100x + 1y10.

(x + y)5 = x5 + 5x4y + 10x3y2 + 10x2y3 + 5xy4 + y5

The first and last coefficients are 1.

The coefficients of the second and second to last terms are equal to n.

1

1

Example 1: What are the last 2 terms of (x + y)10 ?

Since n = 10, the last two terms are 10xy9 + 1y10.

Example: Pascal’s Triangle

Example 3: Use Pascal’s Triangle to expand (2a + b)4.

(2a + b)4 = 1(2a)4 + 4(2a)3b + 6(2a)2b2 + 4(2a)b3 + 1b4

= 1(16a4) + 4(8a3)b + 6(4a2b2) + 4(2a)b3 + b4

= 16a4 + 32a3b + 24a2b2 + 8ab3 + b4

1 1

1st row

1 2 1

2nd row

1 3 3 1

3rd row

1 4 6 4 1

4th row

0th row

1

Extra: different form of questions you might see on test…

• What is the second term in the

binomial expansion of this expression?

(x + 3)4



2. Find and simplify the fourth term in the expansion of (3x2 + y3)4 .

3. (3x - 2y)4 =



[Hint: this is the same as (3x + -2y)4 so use a calculator to find the answer and write the whole expansion for fourth power out for the answer.]