Discover

Consider the binomial x + y. Suppose you want to take this binomial to the nth power: (x + y)n. What is the result of this expansion? Complete Try This 1 to begin exploring this idea.

Try This 1

1. Complete the following table for (x + y)n. Arrange your expanded polynomials from the largest to the smallest exponent on x, as shown for n = 3.
   
   

   n	Binomial Expression (x + y)n	Expansion
   0	(x + y)0
   1	(x + y)1
   2	(x + y)2
   3	(x + y)3	1x3 + 3x2y + 3xy2 + 1y3
   4	(x + y)4
   26	(x + y)26

2. Use patterns in the table to predict the number of terms in the expansion of (x + y)26.

3. Look at the exponent values on x and y.

   1. What is the relationship between the x and y exponents for each term in the expanded form?

   2. Describe a pattern for the x exponents for a given expansion.
   3. Describe a relationship between the n-value and the exponents of the expansion.
4. Use the patterns you described in question 3 to make three predictions about (x + y)26.
5.  
   1. Describe a pattern of the coefficients for a binomial expansion. You may find reviewing the image of Pascal’s triangle from Focus to be helpful.
   2. Can your pattern predict the coefficients for the expansion of (x + y)5? Explain.

Save your responses in your course folder.

Share 1

With a partner or group, discuss the following questions based on the information from Try This 1.

1. Compare the patterns you saw in questions 3 and 5.
2. Do the patterns you have give enough information to determine the first term in the expansion of (x + y)26? What about the second term? What about the rest of the terms?

If required, save a record of your discussion in your course folder.