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APC Unit 2 CH-12.5 Binomial Theorem
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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
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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
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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
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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
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Objective: Student’s will learn how to expand any binomial to any power
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Discovery Activity Expand the following binomials (a) (a + b) 2 = (b) (a+b) 3 = (c) (a+b) 4 =
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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?
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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
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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
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The General Form When to use: When asked for 1 term Or expanding higher powers > 7-8
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You Try…
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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
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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.
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