**High School Number Sense Lesson 55: Properties of Coefficients in a Binomial Expansion**

(I'm not sure why the spacing looks terrible today--I'm sorry!) Today's concept requires a basic understanding of binomials, FOILing, factoring, and Pascal's Triangle. For a great refresher on these topics, please see

**Khan Academy**. This concept appeared

**12 times**last year, with a median placement at

**question # 54**.

**Number Dojo Level: 296**

First, let me introduce (or re-introduce) you to

**Pascal's Triangle**, in which each term in a given row is the sum of the two terms above it. This triangle becomes very useful in many areas of mathematics:

1 1

1 2 1

1 3 3 1

1 4 6 4 1

1 5 10 10 5 1

1 6 15 20 15 6 1

and so on.

For purposes of binomial expansions, the top row (with the lonely 1) is considered

**Row 0**. That makes the (1 1)

**Row 1**, the (1 2 1)

**Row 2**, etc. Also, each row starts with 1, which is the 1st term, etc.

**Finding the Coefficient of the Nth Term of a Binomial Expansion**

The first concept we need to understand is that the

**order**of the binomial expansion is the exponent that the binomial is being raised to. In other words, it will be the highest exponent in the resulting expansion. This order (or highest exponent) corresponds to the relevant row in Pascal's Triangle. So if a binomial is squared, the row we're concerned about is the (1 2 1) row, which is row

**2**.

To determine the coefficient of the nth term, we have to take the original coefficients to the same power as the variables they're attached to, and multiply that by the associated term in the Pascal's Triangle row. This is best seen with examples, so here we go:

**Example 1:**

**The coefficient of the 3rd term of the expansion of (x + y)**

^{4}is ___1. In this expansion, the 1st term will be the

**x**term, the 2nd term will be the

^{4}**x**term, the 3rd term will be the

^{3}y**x**term, and so on.

^{2}y^{2}2. We are concerned about the

**x**term. Notice that each coefficient of the original expansion is

^{2}y^{2}**1**. This makes the problem much easier.

3. The 4th row of Pascal's Triangle has the terms

**1, 4, 6, 4, 1**, and the 3rd term is

**6**. So we will be using a multiplier of 6.

4. Our coefficient will be

**(the coefficient of x)**, which is

^{2}x (the coefficient of y)^{2}x (the multiplier in Pascal's triangle)**1**, which is

^{2}x 1^{2}x 6**1 x 1 x 6**, which is

**6**.

**Example 2:**

**The coefficient of the x**

^{2}y term of (2x + y)^{3}is ___1. In this expansion, the 1st term is

**x**, the 2nd term is

^{3}**x**, and so on.

^{2}y2. The original coefficient of x is

**2**, and the original coefficient of y is

**1**.

3. The 3rd row Pascal's Triangle has the terms

**1, 3, 3, 1**, and the 2nd term is

**3**. So we'll be using a multiplier of 3.

4. Our coefficient will be

**2**, which is

^{2}x 1 x 3**4 x 1 x 3**, which is

**12**.

**Example 3:**

**The sum of the coefficients of the 3rd & 5th terms of (x - y)**

^{5}is ___1. In this expansion, the 1st term is the

**x**, the 2nd term is

^{5}**x**, the 3rd term is

^{4}y**x**, the 4th term is

^{3}y^{2}**x**, the 5th term is

^{2}y^{3}**xy**, and so on.

^{4}2. The original coefficient of x is

**1**, and the original coefficient of y is

**-1**.

3. The 5th row of Pascal's Triangle has the terms

**1, 5, 10, 10, 5, 1**. The 3rd term is

**10**and the 5th term is

**5**.

4. Our 3rd coefficient will be

**1**, which is

^{3}x (-1)^{2}x 10**1 x 1 x 10**, which is

**10**.

5. Our 5th coefficient will be

**1**, which is

^{1}x (-1)^{4}x 5**1 x 1 x 5**, which is

**5**.

6. The sum of

**10**and

**5**is

**15**.

**Now for the Shortcut!**

- To find the
**sum of the coefficients**of a binomial expansion, simply add the coefficients in the expansion and take that sum to the**power (order)**of the expansion.

**Example 4:**

**The sum of the coefficients of (6x + 1)**

^{3}is ___* The sum will be

**(6 + 1)**, which is

^{3}**7**, which is

^{3}**343**.

**Example 5:**

**The sum of the coefficients of (4x - 2y)**

^{5}is ___* The sum will be

**(4 - 2)**, which is

^{5}**2**, which is

^{5}**32**.

**Example 6:**

**The sum of the coefficients of (x - 6y)**

^{4}is ___* The sum will be

**(1 - 6)**, which is

^{4}**(-5)**, which is

^{4}**625**.

**Here's a free worksheet to help you practice ExpandCoeff:**

expandcoeff.pdf |

**Up Next for High School: SolveXFacto**