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The product of some specific binomials can follow certain patterns. These patterns can make calculations easier in contextual situations, such as when calculating a garden's area. This lesson will discuss some of these patterns and how the degree and leading coefficient of the multiplication of polynomials are determined.

### Catch-Up and Review

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

## Seeing Patterns in the Product of Binomials

Calculate the product of the following binomials. Is there a formula to calculate the product of the form What about the product of the form Or the product of the form

## The Square of a Binomial

When a binomial is squared, the resulting expression is a perfect square trinomial.

For simplicity, depending on the sign of the binomial, these two identities can be expressed as one.

### Proof

This rule will be first proven for and then for

This identity can be shown by first rewriting the square as a product.
Multiply parentheses
It has been shown that In this case, when one term of the binomial is subtracted from the other, the middle term of the perfect square trinomial will instead be negative.
Multiply parentheses
It has been shown that

## Using the Square of a Binomial to Represent Areas

Izabella wants to change the decorations in her room. She has two square posters of equal size on her wall that she is thinking of changing. She wants to replace one with a poster that is feet longer on each side than the current poster. The second poster will be replaced by one that is foot smaller on each side. Izabella wants to find an expression in terms of the variable for the difference between the areas. Help her find this expression. Write the answer as a polynomial in standard form.

### Hint

Calculate the area of a square with side lengths feet. Then, calculate the area of a square with side lengths feet. Finally, find the difference between these areas.

### Solution

The area of a square is obtained by squaring its side length.
Consider a square of side length feet. If the side lengths are increased by feet, then the length of the new sides is feet. Therefore, the area of this new square is calculated by squaring To do this, the formula for the square of a binomial can be used.
Simplify right-hand side
Similarly, if the side lengths are decreased by foot, then the length of the new sides is feet. The area of this new square is calculated by squaring Again, the formula for the square of a binomial can be used.
Simplify right-hand side
Finally, to find the difference of the areas in terms of the expression will be subtracted from
The difference of the areas, in terms of is square feet.

## Representing the Area of Praça do Comércio as the Square of a Binomial

The Praça do Comércio is the astonishing main square of Lisbon, the gorgeous capital city of Portugal. Facing the Tagus River, this main court is in the shape of a square with a side length of meters. External credits: Deensel
While researching the city, Izabella wonders about the area of the Praça do Comércio. Find an expression in terms of the variables and for the area of the main square. Write the answer as a polynomial in standard form.

### Hint

The area of a square is calculated by squaring its side length.

### Solution

The area of a square is calculated by squaring its side length.
Therefore, to find the area of the Praça do Comércio, its side length must be squared. Since the expression to be squared is a binomial, the formula for the square of a binomial can be used. The area of the square is square meters.

## Conjugate Binomials

If two binomials differ only in the sign of one of their terms, they are called conjugate binomials.

The binomials and are conjugate binomials.

Here are some examples.

## Product of a Conjugate Pair of Binomials

The product of two conjugate binomials is the difference of two squares.

### Proof

This identity can be proved by using the Distributive Property to multiply the binomials.
Therefore, the product of a binomial and its conjugate is the difference of two squares.

## Area of a Poster

After researching Praça do Comércio, Izabella decided to buy a poster of it to hang in her room. She is deciding between two posters. External credits: Rehman Abubakr, João Eduardo
One poster Izabella is interested in is a square of side length feet. The other is a rectangle of length and width feet. Determine which shape has a greater area.
Calculate the difference between the areas.

### Hint

The area of a square is calculated by squaring its side length. The area of a rectangle is calculated by multiplying the length by the width.

### Solution

To determine which shape has a greater area, both areas will be calculated. The area of a square is calculated by squaring its side length. For the given square, the side length is feet.
The area of a rectangle is calculated by multiplying the length by the width. For the given rectangle, the length and the width are and feet, respectively. These two expressions are conjugate binomials, so the formula for the product of a conjugate pair of binomials can be used.
The area of the rectangle has been found.
Since is greater than the area of the square is greater than the area of the rectangle. Finally, to find the difference, will be subtracted from
The difference between the areas is square feet.

## Leading Coefficient and Degree of a Special Product

When multiplying or squaring binomials, the degree and the leading coefficient of the resulting polynomial may be of interest. Calculate the degree and the leading coefficient of each resulting polynomial.
a
b
c

### Hint

a Use the formula for the square of a binomial.
b Use the formula for the square of a binomial.
c Use the formula for multiplying conjugate binomials.

### Solution

a To find the degree and the leading coefficient of the resulting polynomial, the formula for squaring a binomial will be used.
In the given binomial, and
Simplify
The leading coefficient is and the degree is
b Again, to find the degree and the leading coefficient of the resulting polynomial, the formula for squaring a binomial will be used.
In the given binomial, and
Simplify
The leading coefficient is and the degree is
c In this case, to find the leading coefficient and the degree of the resulting polynomial, two conjugate binomials must be multiplied.
Here, and
Simplify
The leading coefficient is and the degree is

## Finding the Leading Coefficient and Degree of Polynomials

State the degree and the leading coefficient of the resulting polynomial after squaring the binomial or multiplying the conjugate binomials. ## The Cube of a Binomial

The formulas seen in this lesson can be useful to derive other formulas. For example the formula for the square of a binomial can be used to obtain the formula for the cube of a binomial.
Multiplying this equation by will give a rule for the cube of a binomial as it creates a rule for
Simplify right-hand side
To find a rule for can be replaced with in the obtained formula.