4. Special Products of Polynomials
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Chapter 7
4.
Special Products of Polynomials
Understanding special products in algebra, particularly those of binomials, is pivotal for various mathematical applications. These patterns simplify complex calculations, especially in real-world scenarios like determining a garden's area. The lesson sheds light on how the degree and leading coefficient of polynomial multiplication are determined. For instance, when dealing with binomials, certain patterns emerge, making calculations more intuitive. Such knowledge is invaluable for students and professionals alike, aiding in tasks ranging from architectural designs to advanced mathematical research.
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| | 10 Theory slides |
| | 9 Exercises - Grade E - A |
| | Each lesson is meant to take 1-2 classroom sessions |
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.
(a+b)2=a2+2ab+b2(a−b)2=a2−2ab+b2
For simplicity, depending on the sign of the binomial, these two identities can be expressed as one.
(a±b)2=a2±2ab+b2
Proof
This rule will be first proven for (a+b)2 and then for (a−b)2.
(a+b)2=a2+2ab+b2
This identity can be shown by first rewriting the square as a product.(a+b)2
PowToProdTwoFac
a2=a⋅a
(a+b)(a+b)
▼
Multiply parentheses
Distr
Distribute (a+b)
a(a+b)+b(a+b)
Distr
Distribute a
a2+ab+b(a+b)
Distr
Distribute b
a2+ab+ba+b2
CommutativePropMult
Commutative Property of Multiplication
a2+ab+ab+b2
AddTerms
Add terms
a2+2ab+b2
(a−b)2=a2−2ab+b2
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.(a+b)2
PowToProdTwoFac
a2=a⋅a
(a−b)(a−b)
▼
Multiply parentheses
Distr
Distribute (a−b)
a(a−b)−b(a−b)
Distr
Distribute a
a2−ab−b(a−b)
Distr
Distribute -b
a2−ab−ba+b2
CommutativePropMult
Commutative Property of Multiplication
a2−ab−ab+b2
SubTerms
Subtract terms
a2−2ab+b2
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 2 feet longer on each side than the current poster. The second poster will be replaced by one that is 1 foot smaller on each side. 
Izabella wants to find an expression in terms of the variable x 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 x+2 feet. Then, calculate the area of a square with side lengths x−1 feet. Finally, find the difference between these areas.
Solution
The area of a square is obtained by squaring its side length.
Therefore, the area of this new square is calculated by squaring x+2. To do this, the formula for the square of a binomial can be used.
Similarly, if the side lengths are decreased by 1 foot, then the length of the new sides is x−1 feet.
The area of this new square is calculated by squaring x−1. Again, the formula for the square of a binomial can be used.
Finally, to find the difference of the areas in terms of x, the expression x2−2x+1 will be subtracted from x2+4x+4.
The difference of the areas, in terms of x, is 6x+3 square feet.
A=s2
Consider a square of side length x feet. If the side lengths are increased by 2 feet, then the length of the new sides is x+2 feet. A=(x+2)2
ExpandPosPerfectSquare
(a+b)2=a2+2ab+b2
A=x2+2x(2)+22
▼
Simplify right-hand side
CommutativePropMult
Commutative Property of Multiplication
A=x2+2(2)x+22
CalcPow
Calculate power
A=x2+2(2)(x)+4
Multiply
Multiply
A=x2+4x+4
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 3x+y2 meters.
While researching the city, Izabella wonders about the area of the Praça do Comércio. Find an expression in terms of the variables x and y for the area of the main square. Write the answer as a polynomial in standard form.
External credits: Deensel
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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.
The area of the square is 9x2+6xy2+y4 square meters.
A=s2
Therefore, to find the area of the Praça do Comércio, its side length 3x+y2 must be squared. Since the expression to be squared is a binomial, the formula for the square of a binomial can be used.
(3x+y2)2
ExpandPosPerfectSquare
(a+b)2=a2+2ab+b2
(3x)2+2(3x)y2+(y2)2
PowProdII
(ab)m=ambm
9x2+2(3x)y2+(y2)2
PowPow
(am)n=am⋅n
9x2+2(3x)y2+y4
Multiply
Multiply
9x2+6xy2+y4
Discussion
Conjugate Binomials
Rule
Product of a Conjugate Pair of Binomials
The product of two conjugate binomials is the difference of two squares.
(a+b)(a−b)=a2−b2
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.
(a+b)(a−b)
Distr
Distribute (a−b)
a(a−b)+b(a−b)
Distr
Distribute a
a2−ab+b(a−b)
Distr
Distribute b
a2−ab+ba−b2
CommutativePropMult
Commutative Property of Multiplication
a2−ab+ab−b2
AddTerms
Add terms
a2−b2
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.
One poster Izabella is interested in is a square of side length x feet. The other is a rectangle of length x+2 and width x−2 feet. Determine which shape has a greater area. Calculate the difference between the areas.
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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 x feet.
Area of the Squarex2 ft2
The area of a rectangle is calculated by multiplying the length by the width. For the given rectangle, the length and the width are x+2 and x−2 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.
Area of the Rectanglex2−4 ft2
Since x2 is greater than x2−4, the area of the square is greater than the area of the rectangle. Finally, to find the difference, x2−4 will be subtracted from x2.
The difference between the areas is 4 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 (3x3+x2)2
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b (x5−5x)2
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c (2x3+1)(2x3−1)
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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.
The leading coefficient is 9 and the degree is 6.
(a+b)2=a2+2ab+b2
In the given binomial, a=3x3 and b=x2.
(3x3+x2)2
▼
Simplify
ExpandPosPerfectSquare
(a+b)2=a2+2ab+b2
(3x3)2+2(3x3)x2+(x2)2
PowProdII
(ab)m=ambm
9(x3)2+2(3x3)x2+(x2)2
PowPow
(am)n=am⋅n
9x6+2(3x3)x2+x4
Multiply
Multiply
9x6+6x3x2+x4
MultPow
am⋅an=am+n
9x6+6x5+x4
b Again, to find the degree and the leading coefficient of the resulting polynomial, the formula for squaring a binomial will be used.
The leading coefficient is 1 and the degree is 10.
(a−b)2=a2−2ab+b2
In the given binomial, a=x5 and b=5x.
(x5−5x)2
▼
Simplify
ExpandNegPerfectSquare
(a−b)2=a2−2ab+b2
(x5)2−2x5(5x)+(5x)2
PowPow
(am)n=am⋅n
x10−2x5(5x)+(5x)2
CommutativePropMult
Commutative Property of Multiplication
x10−2(5x)x5+(5x)2
Multiply
Multiply
x10−10x⋅x5+(5x)2
MultBasePow
a⋅am=a1+m
x10−10x6+(5x)2
PowProdII
(ab)m=ambm
x10−10x6+25x2
IdPropMult
Identity Property of Multiplication
1x10−10x6+25x2
c In this case, to find the leading coefficient and the degree of the resulting polynomial, two conjugate binomials must be multiplied.
The leading coefficient is 4 and the degree is 6.
(a+b)(a−b)=a2−b2
Here, a=2x3 and b=1.
(2x3+1)(2x3−1)
▼
Simplify
ExpandDiffSquares
(a+b)(a−b)=a2−b2
(2x3)2−12
PowProdII
(ab)m=ambm
4(x3)2−12
PowPow
(am)n=am⋅n
4x6−12
BaseOne
1a=1
4x6−1
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.
To find a rule for (a−b)3, b can be replaced with -b 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.
(a+b)2=a2+2ab+b2
Multiplying this equation by (a+b) will give a rule for the cube of a binomial as it creates a rule for (a+b)3.
(a+b)2=a2+2ab+b2
MultEqn
LHS⋅(a+b)=RHS⋅(a+b)
(a+b)2(a+b)=(a2+2ab+b2)(a+b)
CommutativePropMult
Commutative Property of Multiplication
(a+b)(a+b)2=(a2+2ab+b2)(a+b)
MultBasePow
a⋅am=a1+m
(a+b)3=(a2+2ab+b2)(a+b)
(a+b)3=a3+3a2b+3ab2+b3
(a+b)3=a3+3a2b+3ab2+b3
SubstituteExpressions
Substitute expressions
(a+(-b))3=a3+3a2(-b)+3a(-b)2+(-b)3
▼
Simplify
MultPosNeg
a(-b)=-a⋅b
(a+(-b))3=a3+(-3a2b)+3a(-b)2+(-b)3
NegBaseToNegPow
(-a)3=-a3
(a+(-b))3=a3+(-3a2b)+3a(-b)2+(-b3)
AddNeg
a+(-b)=a−b
(a−b)3=a3−3a2b+3a(-b)2−b3
NegBaseToPosPow
(-a)2=a2
(a−b)3=a3−3a2b+3ab2−b3
(a+b)3=a3+3a2b+3ab2+b3(a−b)3=a3−3a2b+3ab2−b3⇓(a±b)3=a3±3a2b+3ab2±b3
Simplify each expression.
(x+2)(x−2)
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(3x+7)(3x−7)+49
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(x+5)2+(x−5)2
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Notice that the factors of the given product are two conjugate binomials. Therefore, we will use the formula for the product of conjugate binomials. (a+b)(a-b)=a^2-b^2 To do so, keep in mind that the square of a square root is equal to its radicand. Let's multiply the binomials!
We will apply the formula for the product of conjugate binomials again. Let's do it!
This time we will use the formula for the square of a binomial. Let's recall the formulas for both a sum and a difference of binomials. (a+b)^2=a^2+2ab+b^2 (a-b)^2=a^2-2ab+b^2 With this information in mind, we can simplify the given expression.
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The figures below are squares.
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To find the area of the shaded region, we need to find the area of the larger square, then subtract the area of the smaller square. Let A_R be the area of the shaded region, A_L the area of the larger square, and A_S the area of the smaller square. A_R=A_L-A_S Recall that the area of a square is found by squaring its side length. A=s^2 In this formula, s is the side length of the square. We can calculate the area of the larger square A_L by substituting x+1 into the formula.
We will now calculate the area of the smaller square A_S. Let's substitute its side length x-3 into the formula.
Now that we have found A_L and A_S, we can calculate A_R, the area of the shaded region.
The area of the shaded region is 8x-8 square units.
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Simplify the following expression.
(a−2b+3c)2
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We will simplify the expression (a-2b+3c)^2. Notice that this is no longer a binomial. Therefore, we cannot use the square of binomial rules to simplify this expression.
This is as far as the expression can be simplified.
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Special Products of Polynomials
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