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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.
Challenge
Seeing Patterns in the Product of Binomials
Discussion
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
Distr
Distribute
Distr
Distribute
Distr
Distribute
CommutativePropMult
Commutative Property of Multiplication
AddTerms
Add terms
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
Distr
Distribute
Distr
Distribute
Distr
Distribute
CommutativePropMult
Commutative Property of Multiplication
SubTerms
Subtract terms
Example
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.
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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.
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.
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.
▼
Simplify right-hand side
▼
Simplify right-hand side
Example
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.
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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. {"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":false,"useShortLog":false,"variables":["x","y"],"constants":[]},"rendered":false},"text":""},"formTextBefore":"Area:","formTextAfter":"square meters","answer":{"text":["9x^2+6xy^2+y^4","9x^2+6y^2x+y^4","y^4+6xy^2+9x^2","y^4+6y^2x+9x^2"]}}
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.
Discussion
Conjugate Binomials
Rule
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.
Distr
Distribute
Distr
Distribute
Distr
Distribute
CommutativePropMult
Commutative Property of Multiplication
AddTerms
Add terms
Example
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.
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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. {"type":"choice","form":{"alts":["Square","Rectangle"],"noSort":false},"formTextBefore":"","formTextAfter":"","answer":0}
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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.
Example
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
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b
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":[],"constants":[]},"rendered":false},"text":""},"formTextBefore":"Degree:","formTextAfter":null,"answer":{"text":["10"]}}
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c
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":[],"constants":[]},"rendered":false},"text":""},"formTextBefore":"Degree:","formTextAfter":null,"answer":{"text":["6"]}}
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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
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
The leading coefficient is and the degree is
▼
Simplify
CommutativePropMult
Commutative Property of Multiplication
Multiply
Multiply
IdPropMult
Identity Property of Multiplication
c In this case, to find the leading coefficient and the degree of the resulting polynomial, two conjugate binomials must be multiplied.
Here, and
The leading coefficient is and the degree is
Pop Quiz
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.
Closure
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
To find a rule for can be replaced with in the obtained formula.
By using the formula for the square of a binomial, two formulas for the cube of a binomial were derived. For simplicity, depending on the sign of the binomial, these two identities can be expressed as one.
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