Ms. Nong Digital Lesson (Play the presentation and turn on your volume)

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..?

Fill in the missing numbers for this triangle on your paper

2 0 = = 1+1 = = = = = = = 16 The Sums of the Rows The sum of the numbers in any row is equal to 2 to the n th power or 2 n, 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 2 n 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 Can you guess the next row?

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.

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 9 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. Patterns observed Consider the patterns formed by expanding (x + y) n. P owers 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 = x 2 + 2xy + y 2 (x + y) 3 = x 3 + 3x 2 y + 3xy 2 + y 3 (x + y) 4 = x 4 + 4x 3 y + 6x 2 y 2 + 4xy 3 + y 4 (x + y) 5 = x 5 + 5x 4 y + 10x 3 y x 2 y 3 + 5xy 4 + y 5 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 4 th term of (x + y) 5 is a term with x 2 y 3.”

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 10 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 + 1y 10. Use a calculator to calculate (2) 9 = 510 then multiply it by 10 Your final answer should be 5100x + 1y 10. (x + y) 5 = x 5 + 5x 4 y + 10x 3 y x 2 y 3 + 5xy 4 + y 5 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 10xy 9 + 1y 10.

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 11 Example 3: Use Pascal’s Triangle to expand (2a + b) 4. (2a + b) 4 = 1(2a) 4 + 4(2a) 3 b + 6(2a) 2 b 2 + 4(2a)b 3 + 1b 4 = 1(16a 4 ) + 4(8a 3 )b + 6(4a 2 b 2 ) + 4(2a)b 3 + b 4 = 16a a 3 b + 24a 2 b 2 + 8ab 3 + b 4 Example: Pascal’s Triangle 1 1 st row nd row rd row th row 0 th row1

Extra: different form of questions you might see on test… 1.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 (3x 2 + y 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.]