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Expanding binomials can be difficult if the exponent is large. Considering that there is a formula for the square of a binomial, it might be possible to find similar expressions for greater exponents. This lesson aims to show an array of numbers that can be used to expand binomials.

### Catch-Up and Review

Here are a few recommended readings before getting started with this lesson.

## Recognizing Patterns

In the following diagram, an incomplete triangle partially filled with numbers is displayed. Complete the triangle by inserting the numbers listed at the bottom into their correct place within the triangle. Considering the given numbers, does the triangle follow a pattern? If so, what is it?

## Difference of Squares

Two good friends, Diego and Maya, like to challenge each other on math related topics. One summer afternoon, Diego came up with an interesting problem. Help Maya prove that the difference of the squares of any two consecutive integers is odd.

See solution.

### Hint

Let be an arbitrary integer. Find the difference of the squares of and its consecutive integer.

Remember that the division of an even number by two always result on an integer.

### Solution

To prove that the difference of the squares of any two consecutive integers is odd, let be an integer number. The second of the consecutive integers is obtained by adding to Therefore, the consecutive integer to is Each number will be squared.
Next, the difference of the squared numbers can be calculated. For simplicity the smaller integer is subtracted from the larger, but it should be noted that either way works.
Simplify
To determine whether this difference is an odd number, the expression can be divided by If the result is not an integer, the difference is odd.
Simplify
Since is an integer number and is not, the number is not an integer. This means that the number is odd for any integer Therefore, the difference of the squares of any two consecutive integers odd.

## Difference of Cubes

After solving Diego's challenge, Maya now creates one for Diego. She asks him to prove that the difference of the cubes of any two consecutive integers is also an odd number. Help Diego prove this statement!

See solution.

### Hint

To prove this statement, consider an even number where is any integer. Its consecutive number is and its previous number is Consider the difference of the cubes of and and the difference of and

### Solution

Any even number can be written as where is an integer. Consider the cube of this number.
Now, when considering two consecutive integers, one is even and the other is odd. The odd number can be before or after These cases can be expressed by subtracting from or adding to respectively.
Recall the formula for the cube of a binomial.
By using this formula, the cubes of and can be found. These formulas are obtained by substituting for and for
Simplify
There are two possible situations. The consecutive numbers can be and or and These two situations will be considered one at a time.

### and

If the consecutive integers are and then their cubes are and respectively. Next, find their difference. For the result to be positive, the cube of is subtracted from the cube of
Simplify
Now, divide this expression by An even number is always divisible by two. Therefore, if dividing the above expression by 2 results in an integer, the expression represents an even number.
Simplify
This expression will be examined closely to determine if it is an integer or not. Each term will be examined individually, gradually building the expression of
Expression Is it an Integer? Reason
Yes It is defined this way.
Yes Closure Property of Multiplication
Yes Closure Property of Multiplication
Yes Closure Property of Multiplication
Yes Closure Property of Subtraction
No is not an integer

Since dividing by did not result in an integer number, the expression which is the difference of the cubes of and is odd.

### and

If the consecutive integers are and then their cubes are and respectively. Their difference can be calculated.
Simplify
Just like in the previous case, the expression that represents the difference will be divided by to determine if the result is even.
Simplify
Similar to the previous case, is an integer. Because of the the sum is not an integer. Therefore, the difference of the cubes is odd. Now that the two cases are considered, the statement has been proved.

## Square of the Sum of Two Consecutive Integers

Diego, engulfed by Maya's last challenge, asks if she could create another one for him. This time, Maya asks Diego to prove that the square of the sum of two consecutive integers is odd. Help Diego solve Maya's challenge!

See solution.

### Hint

Let and be two consecutive integers. Find their sum, calculate its square, and prove that the obtained result is odd.

Remember that the division of an even number by two always result on an integer.

### Solution

Let and be two consecutive integer numbers. To prove the given statement, the numbers first must be added.
The sum resulted in a binomial. Then, the square of the sum is the square of a binomial. An even number is always divisible by This means that, if dividing the above expression by results in an integer, the expression represents an even number. Otherwise, it represents an odd number.
Since is an integer, the expression also represents an integer. Adding a non-integer to an integer results in a non-integer. Therefore, the expression is not an integer. This means that is an odd number, proving the statement.

## Pascal's Triangle and the Binomial Theorem

The Pascal's triangle is a triangular representation of the coefficients in the expansion of a binomial expression. In other words, it contains the coefficients that result when expanding the expression with The rows are numbered from top to bottom starting with The number in each cell equals the sum of the numbers in the two neighboring cells above. The numbers in the Pascal's Triangle have an interesting application. Consider the following binomial expansions. Looking at the coefficients of the binomial expansions, it can be noted that every number of a row in the Pascal's Triangle is the same as the coefficients of a binomial expansion. This is expressed in the Binomial Theorem.

### Binomial Theorem

Consider a binomial raised to the power of where is a positive integer. Let and be real numbers. Expanding the binomial results in the following expression. In this expansion, the numbers are the numbers in the th row of Pascal's Triangle.

## The Volume of a Room

Maya, feeling like she challenged Diego well, remembers that her mom asked her to measure her room for new wallpaper. She has to take a phone call, so she asks Diego if he can take the measurements. The room is in the shape of a cube. Diego hands over the measurements to her — he wrote them as a challenge! Diego wrote that each side of the room has a length of Write the expanded polynomial for the volume of the room.

### Hint

Use the Binomial Theorem to expand the binomial.

### Solution

The room has the shape of a room with side length To find its volume, it is necessary to use the formula for the volume of a cube.
In this formula, is the volume of the cube and its side length. Substituting for gives the formula for the volume of the room. Instead of multiplying the binomial by itself times, the Binomial Theorem can be used to expand the binomial. This theorem indicates that the coefficients of a binomial to the sixth power coincide with the numbers in the of Pascal's Triangle. Each number in the sixth row corresponds to a coefficient of the binomial expansion, considering that the result is written in standard form.
Simplify right-hand side

## Finding a Certain Coefficient

It's Diego's birthday, and without fail, Maya has gotten him a cool present. Diego is so excited to open it, but there is a problem. The gift has a lock that asks for a combination that he does not have! Maya, of course, wants to have a bit of fun and tells Diego that the combination is the fifth term in the expansion Help Diego open his gift!

### Hint

Consider the Binomial Theorem.

### Solution

The Binomial Theorem indicates that the coefficients in the expansion of coincide with the numbers in the ninth row of Pascal's Triangle. The fifth term in the expansion of the binomial raised to the ninth power matches the fifth number in the ninth row of Pascal's Triangle. The terms of the polynomial are counted starting from the term this being the first one. The variables for each consecutive term are obtained by subtracting one from the exponent of and adding one to the exponent of The coefficients are the same as the numbers in the ninth row of Pascal's Triangle.

term
Number of Term Coefficient Variables Value of Term

Therefore, the fifth term is

## Using the Binomial Theorem to Solve a Problem

To finish this lesson, a more complex exercise will be solved using the Binomial Theorem. Consider the following expression.
It is known that the coefficient in the term in the expansion of the above expression is the same as the coefficient in the term. Find the possible values of Write the exact answer in its simplest form, with no radicals in denominators.

### Hint

Use the Binomial Theorem to find the coefficients of every term.

### Solution

The Binomial Theorem indicates that the coefficients in the expansion of are the numbers in the sixth row of Pascal's Triangle. Now the terms for and have to be identified. To identify these terms, every term of the binomial expansion is determined using the Binomial Theorem.

Number of Term Coefficient Variables Value of Term
It is given that the coefficients of the and the terms are the same. Therefore, must be equal to This can be written as an equation which can be solved for
Solve for

Simplify right-hand side
Therefore, the possible values for are and
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