Patterns in Binomial Expansion 1 Assessment Task: 1) Complete the following tasks and questions looking for any patterns. Show all your work! Write neatly in the space provided. 2) Write a rule or formula in mathematical language to assist in extending a problem. IMPORTANT Vocabulary: 1) Term a product of numbers and variables Example: 4x 2 is a term. So is 3x or 5xy or ½x 2 y 4. But 5x 2y is not a term. It is a sum of terms, the second of which is 2y. 2) Coefficient the number multiplying a variable (generally in front of the variable) Example: 4x 2 3x+ 2 4, -3, and 2 are all coefficients. 2 is special because there is no variable beside it, it can also be called a constant (no change in value). Notice that the 3 is negative coefficients include the sign. 3) Expand and Simplify apply distributive property to the expressions, remembering that parentheses are used for multiplying. Like terms are combined. Example: (x + 2)(x + 1) = x 2 + x + 2x + 2 = x 2 + 3x + 2 By expanding, we produce 4 terms. However the expression is not simplified By combining x and 2x into 3x, the expression is now simplified There are several ways to understand multiplying sums of terms. The fundamental idea is that each term in one set of parentheses needs to be multiplied by each term in the other set of parentheses. The results are then added together (with attention to signs). FOIL (First Outer Inner Last), Happy Man good for two binomials (an expression with two terms) Rainbow, Paper Boy use arrows to keep track. Good for many terms. Grid Use a grid to keep track. Very good for 3 or more terms. Helps really see what s happening. 4) Descending Order ordering terms in an expression by exponent size from largest to smallest from left to right Example: 4x 2 3x + 2, rather than 2 + 4x 2 3x (even though the two expressions are equivalent) 1 Adapted from Thauvette, Simon M., MYP 5 Workshop, Sep 13-15, 2013, Hong Kong, http://aprilmypworkshop.wikispaces.com/file/detail/year+3+patterns+in+binomial+expansion+investigation.doc C:\Users\Bob\Documents\Dropbox\Desert\SL1\1 - Algebra&Functions\BinomialExpansionInvestigation.docx on 12/26/17 at 11:45 AM Page 1 of 9
PART I: Producing Data through Expansion Example A) Expand and simplify (x + 1) 1 The binomial in example (A) has an exponent of? How many terms are there in the simplified answer to (A)? What are the coefficients, in order, for each term in the simplified answer to (A)? Example B) Expand and simplify (x + 1) 2 The binomial in example (B) has an exponent of? How many terms are there in the simplified answer to (B)? What are the coefficients, in order, for each term in the simplified answer to (B)? Example C) Expand and simplify (x + 1) 3 The binomial in example (C) has an exponent of? How many terms are there in the simplified answer to (C)? What are the coefficients, in order, for each term in the simplified answer to (C)? C:\Users\Bob\Documents\Dropbox\Desert\SL1\1 - Algebra&Functions\BinomialExpansionInvestigation.docx on 12/26/17 at 11:45 AM Page 2 of 9
Example D) Expand and simplify (x + 1) 4 The binomial in example (D) has an exponent of? How many terms are there in the simplified answer to (D)? What are the coefficients, in order, for each term in the simplified answer to (D)? C:\Users\Bob\Documents\Dropbox\Desert\SL1\1 - Algebra&Functions\BinomialExpansionInvestigation.docx on 12/26/17 at 11:45 AM Page 3 of 9
PART II: Organizing Data Instructions: 1. The first row of squares represents example (A). Write the coefficients, in order, in the first row. Identify the exponent for example (A); write it beside the first square on the first row. 2. The second row of squares represents example (B). Write the coefficients, in order, in the second row. Identify the exponent for example (B); write it beside the first square on the second row. 3. The third row of squares represents example (C). Write the coefficients, in order, in the third row. Identify the exponent for example (C); write it beside the first square on the third row. 4. The fourth row of squares represents example (D). Write the coefficients, in order, in the fourth row. Identify the exponent for example (D); write it beside the first square on the fourth row. BUT WHAT ABOUT THE EMPTY ROWS?! C:\Users\Bob\Documents\Dropbox\Desert\SL1\1 - Algebra&Functions\BinomialExpansionInvestigation.docx on 12/26/17 at 11:45 AM Page 4 of 9
PART III: Describing the Pattern Q1: Can you find a pattern above that relates the numbers in the rows to each other? If so, describe in words how each row relates to the row before and after it. Please be thorough in your explanation. You may use examples or a diagram to assist your explanation. Q2: Once you understand the pattern, and have described it, use your pattern to fill in all of the rows with the correct numbers. C:\Users\Bob\Documents\Dropbox\Desert\SL1\1 - Algebra&Functions\BinomialExpansionInvestigation.docx on 12/26/17 at 11:45 AM Page 5 of 9
Q3: a) What is the relationship between the row number and the exponent in the expansion? Describe using words. b) What is the relationship between the numbers you entered in each row and the terms in each expansion? Describe. C:\Users\Bob\Documents\Dropbox\Desert\SL1\1 - Algebra&Functions\BinomialExpansionInvestigation.docx on 12/26/17 at 11:45 AM Page 6 of 9
PART IV: General Rule, Application, and Testing of Pattern Using the pattern you found and without doing any multiplying, expand the following: Example E) (x + 1) 5 Example F) (x + 1) 6 Example G) (x + 1) 7 Q4: Consider your answers to (E), (F), and (G). Write down a GENERAL RULE that describes the patterns in the triangle and how exactly you would use it to expand binomials. Justify how you created this rule. C:\Users\Bob\Documents\Dropbox\Desert\SL1\1 - Algebra&Functions\BinomialExpansionInvestigation.docx on 12/26/17 at 11:45 AM Page 7 of 9
PART V: Reflection and Evaluation Questions 1) How could these patterns help you with your homework and assessments? 2) The top row in the diagram of question Q2 has 2 elements in it. If you wanted to complete the triangle with another row above that, having only one element, what would that element be? What would the label to the left of it be? How can you interpret the number in this row as an expansion of a binomial? 3) How would the pattern change if the binomial were (x 1) 2 instead of (x + 1) 2? Why? 4) How would the pattern change if the binomial were (x 1) 3 instead of (x + 1) 3? Why? 5) Explain how you would have to change your general rule from Q4 above to work for (x 1) n? C:\Users\Bob\Documents\Dropbox\Desert\SL1\1 - Algebra&Functions\BinomialExpansionInvestigation.docx on 12/26/17 at 11:45 AM Page 8 of 9
6) Suppose the binomial being expanded were more general. Explain how you would have to change your general rule from Q4 to work for (a + b) n? C:\Users\Bob\Documents\Dropbox\Desert\SL1\1 - Algebra&Functions\BinomialExpansionInvestigation.docx on 12/26/17 at 11:45 AM Page 9 of 9