APC Unit 2 CH-12.5 Binomial Theorem

Warm-up  Take your Homework out  Clearly Label 12.2, 12.3, and 12.4  Ask your Questions  While I’m Checking… Complete the Homework Check-in worksheet

Series Proof  Step 1: Show that n=1 creates a true statement  Step 2: Assume the given statement is valid (true)  Step 3: Show that n=K+1 is true  Write out the series including the k term and the k+1 term (on the left side)  Substitute (k+1) for n (on the right side)  Group the 1 through K terms and substitute the given statement  Simplify the Left side to match the Right side  Step 4: State your conclusion  Example 1 together  Page 836 #1 on your own

Inequality Proof  Step 1: Show that n=1 creates a true statement  Step 2: Assume the given statement is valid (true)  Step 3: Show that n=K+1 is true  Write the inequality using n=K+1  Substitute (k+1) for n (on the right side)  Factor out the original term using exponent rules  Add a clause to the inequality using the original inequality  Show that the inequalities are true  Step 4: State your conclusion  Example 2 together

Divisibility Proofs  Step 1: Show that n=1 creates a true statement  Step 2: Assume the given statement is valid (true)  Write the statement as a multiple of the number  Step 3: Show that n=K+1 is true  Substitute (k+1) for n  Factor out the original term using exponent rules  Re-arrange the statement and substitute  Distribute and then factor  Step 4: State your conclusion  Example 4 together  Example 5 on your own

Objective:  Student’s will learn how to expand any binomial to any power

Discovery Activity  Expand the following binomials  (a) (a + b) 2 =  (b) (a+b) 3 =  (c) (a+b) 4 =

Write the Coefficients in the form of a triangle  What does it look like?  What patterns do you see?  Can you predict the coefficients for the next exponent?

Pascal’s Triangle  Use it if you like  When to use it?  Expansions less than 7 or 8  When not to use it?  When asked to find one coefficient

Writing the general Coefficient Algebraically  n C r – the combination of n taken r at a time  n = the power of the expansion (or row of the triangle)  r = the term within the expansion (starting with a 0 term)  Notation

The General Form  When to use:  When asked for 1 term  Or expanding higher powers > 7-8

You Try…

n C r a n−r b r Remember : n = the exponent of the expression r = the exponent of the second part of the binomial The exponent of the first part plus the exponent of the second part = n

Easier to just try it  r = the term minus 1  Try Worksheet 5, 6, and 7  Hint: #5 n = 10, r=3  Hint: #6 the sixth term is r = 5  Hint: #7 The constant term the exponents add up to zero.