# GRE Subject Test: Math : Binomial Expansion

## Example Questions

### Example Question #1 : Classifying Algebraic Functions

Expand: .

Explanation:

Step 1: Evaluate .

Step 2. Evaluate

From the previous step, we already know what  is.

is just multiplying by another

Step 3: Evaluate

The expansion of  is

### Example Question #2 : Classifying Algebraic Functions

What is the expansion of ?

Explanation:

Solution:

We can look at Pascal's Triangle, which is a quick way to do Binomial Expansion. We read each row (across, left to right)

For the first row, we only have a constant.
For the second row, we get .
...
For the 7th row, we will start with an  term and end with a constant.

Step 1: We need to locate the 7th row of the triangle and write the numbers in that row out.

The 7th row is .

Step 2: If we translate the 7th row into an equation, we get:

. This is the solution.

### Example Question #1 : Binomial Expansion

Expand:

Explanation:

Method One:

We will start expanding slowly, and we will end up at exponent

Step 1: Expand:

Step 2: Multiply  by the product of . By doing this, we are now expanding .

Step 3: Multiply by  again

.

After Step 4:
After Step 5:

After Step 6, the final answer is:

.

Method Two:

You can find the expansion of this binomial by using the Pascal's Triangle (shown below)

If you look at Row  of the triangle above, the row that starts with .

We need to negate every nd term, as the answer in Method One has every even term negative.

We will still get the answer: .

### Example Question #1 : Binomial Expansion

Expand:

Explanation:

Step 1: Expand

Step 2: FOIL the first two parentheses:

Step 3: Multiply the expansion in step 2 by :

The expanded form of  is .

### Example Question #3 : Binomial Expansion

Expand:

Explanation:

Step 1: Multiply

Step 2: Multiply the result in step 1 by

Step 3: Multiply the result of step 2 by

Simplify:

Explanation:

### Example Question #11 : Algebra

Explanation:

The easiest way to expand binomials raised to higher powers is to use Pascal's Triangle.

Pascal's Triangle is used to find the multipliers for each level of exponent.

It follows the pattern listed below.

To complete the expansion we will take the row that corresponds to the 4th exponent for this problem.

We will now organize this into columns and rows.

The second and third rows are organized by taking the left term from the highest to lowest power, and the right term from lowest to highest power.

Since the x is on the left, it is raised to the 4th power, 3rd power and so on.

Since the 3 is on the right, it is raised to the 0 power, 1 power and so on.

We now simplify each of the terms, the bottom row is the only one to be simplified in this case. Anything to the zero power except zero is 1.

Now we multiply each column together to obtain the full expansion.

For example to obtain the third term we multiplied everything in the 3rd column:

We did this for all of the columns to get the below final answer.

### Example Question #1 : Binomial Expansion

Expand:

None

Explanation:

We multiply the expansion of  by :

Multiply again by :

Multiply by :

### Example Question #11 : Polynomials

Expand.

Explanation:

Expand by distributing each of the factors

Simplify

Simplify

### Example Question #8 : Binomial Expansion

Expand:

None

Explanation:

Step 1: Let's start small, expand .

Step 2: Expand

Take the final answer of Step 1 and multiply it by ...

Step 3: Multiply again by  to the final answer of Step 2...