Your task as Agent Binomial Expansion, should you choose to accept it, is to expand the binomial

### Pascal's Triangle

Each row begins and ends with a one. Each “interior” value in each row is the sum of the two numbers above it. For example,

Another pattern that can be observed is that the row number is equal to the number of elements in that row. Row 1, for example has 1 element, 1. Row 2 has 2 elements, 1 and 1. Row 3 has 3 elements, 1, 2 and 1.

A third pattern is that the second element in the row is equal to one less than the row number. For example, in row 5 we have 1, 4, 6, 4 and 1.

Let's continue the triangle to determine the elements in the

Following the pattern of adding adjacent elements to get the elements in the next row, we find hat the eighth row is:

Now, continue the pattern again to find the

Now, let's expand the binomial

- Take two binomials at a time and square them using
(a+b)2=a2+2ab+b2 - Next, distribute each term in the first trinomial over each term in the second trinomial and collect like terms.

We can see that the powers of

Finally, let's use what was discovered in the previous problem to expand

The degree of this expansion is 6, so the powers of

In the previous problem we observed that the coefficients for a fourth degree binomial were found in the fifth row of Pascal’s Triangle. Here we have a

### Examples

#### Example 1

Earlier, you were asked to expand the binomial

To expand the binomial

The degree of this expansion is 5, so the powers of

The coefficients for a fifth degree binomial can be found in the sixth row of Pascal’s Triangle. Now we can fill in the blanks with the correct coefficients, replacing *y* with

#### Example 2

Write out the elements in row 10 of Pascal’s Triangle.

We continued the triangle to find the

Subsequently, the

#### Example 3

Expand

#### Example 4

Write out the coefficients in the expansion of

### Review

- Write out the elements in row 7 of Pascal’s Triangle.
- Write out the elements in row 13 of Pascal’s Triangle.

Use Pascal’s Triangle to expand the following binomials.

(x−6)4 (2x+5)6 (3−x)7 (x2−2)3 (x+4)5 (2−x3)4 (a−b)6 (x+1)10

### Answers for Review Problems

To see the Review answers, open this PDF file and look for section 12.6.