Explore

 

In Try This 1 you may have noticed that expanding a binomial power by multiplying is a very tedious task, even for relatively small exponents. There are a variety of patterns that occur when you expand the expression (x + y)n. Some of these patterns include the following:

• The number of terms in the expansion is equal to n + 1. For example, (x + y)15 will have 16 terms.
• The exponent of x and the exponent of y in each term add up to n. For example, if n = 3, each pair of exponents in a term adds up to 3.
  
  
   
  (x + y)3 = 1x3y0 + 3x2y1 + 3x1y2 + 1x0y3
• The exponents on the x in each term in the expansion will include n, n − 1, …, 2, 1, 0. For example, if n = 4
  
  
   
  (x + y)4 = 1x4y0 + 4x3y1 + 6x2y2 + 4x1y3 + 1x0y4
• The exponents on the y in each term in the expansion will include n, n − 1, …, 2, 1, 0. For example, if n = 4
  
  
   
  (x + y)4 = 1x4y0 + 4x3y1 + 6x2y2 + 4x1y3 + 1x0y4

The rest of this lesson focuses on these patterns, which help to make expanding a binomial easier. The pattern for the coefficients of the terms is a bit more elusive. You will explore this pattern using Pascal’s triangle in Try This 2.

 

Try This 2

 

The first five rows of Pascal’s triangle are shown.

 

1. Examine the triangle. Describe at least three patterns you see.
2. Determine rows 6 and 7.
3. Recall the binomial powers that you expanded in Try This 1.
   
   
    
   
   1. Describe the relationship between the binomial expansions and Pascal’s triangle.
   2. Use Pascal’s triangle to predict the coefficients in the expansion of (x + y)6.
   3. Complete the expansion of (x + y)6.
   4. Predict the first four terms of the expansion of (x + y)10.

Save your responses in your course folder.

 

Share 2

 

With a partner or group, discuss the following questions based on the patterns you determined in Try This 2.

1. Compare the patterns you found in Pascal’s triangle. How did your strategy to determine the next rows compare with others?
2. Compare the relationships you determined between Pascal’s triangle and binomial expansions.
3. Explain why the strategy of writing out the rows of Pascal’s triangle may not be reasonable to determine the first four terms of (x + y)33.

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