Topics
6. Polynomials & Factoring
topic badge
United States of AmericaVirginia
Algebra 1

6.02 Multiply polynomials

Lesson
Practice

Multiply polynomials

Exploration

Complete the area models for multiplication shown:

A rectangle divided into 2 rectangles. The side next to the left rectangle is labeled 3 while the top is labeled 7. The top of the right rectangle is labeled 9.
3(7+9)
A rectangle divided into 2 rectangles. The side next to the left rectangle is labeled 3 x squared while the top is labeled 7 x cubed. The top of the right rectangle is labeled 9.
3x^2(7x^3+9)

Complete the new area models for multiplication:

A rectangle divided into 2 rows of 2 rectangles. The outside of the rectangle is labeled 20 above the left column and 1 above the right column. The side is labeled 10 next to the top row and negative 3 next to the bottom row.
(10-3)(20+1)
A rectangle divided into 2 rows of 2 rectangles. The outside of the rectangle is labeled 2 x above the left column, 1 above the right column, and a plus sign above the line between the left and right column. The side is labeled x next to the top row and negative 3 next to the bottom row.
\left(x-3\right)\left(2x+1\right)

  1. What do the area models have in common?

  2. What's different about the area models?

  3. Make a conjecture about how multiplying polynomials relates to multiplying integers.

Consider the garden plot:

A garden plot with length of 4x+1 and width of 3x-2

The length of a rectangular garden plot is 4x+1 feet. The width of the plot is 3x - 2 feet. The area can be represented as the product \left(4x+1\right)\left(3x-2\right). We can use models to simplify the product as a polynomial.

An area model using tiles to show 4x+1 multiplied by 3x-2. Ask your teacher for more information.
An algebra tiles model
An area model using tiles to show 4x+1 multiplied by 3x-2. Ask your teacher for more information.
A box/area model

Notice the algebra tiles model shows each individual tile, but the box model combines some like terms together. Both models show the product of \left(4x+1\right)\left(3x-2\right)=12x^{2}+3x-8x-2. Which we can simplify by combining like terms to 12x^{2}-5x-2.

An area model using tiles to show a+b multiplied by c+d. Ask your teacher for more information.

\,\\\,The distributive property can be used to multiply two polynomials:(a+b)\left(c+d\right)=ac + ad + bc + bd

Area models help us visualize the different terms from the distributive property. They can help us organize the multiplication of polynomials, so we don't miss any terms. Then, we can combine like terms to get the simplest polynomial.

The product of two polynomials will always result in a new polynomial where

  • The degree of the new polynomial will be the sum of the degrees of the multiplied polynomials.

  • The number of terms may vary from the original polynomials depending on how like terms are combined.

Examples

Example 1

Multiply 3 x \left( 2 x^{2} - 5 x + 4\right).

Worked Solution
Apply the idea

We can use the distributive property to get the product of the monomial 3 x and the trinomial 2 x^{2} - 5 x + 4.

\displaystyle 3 x \left( 2 x^{2} - 5 x + 4\right)\displaystyle =\displaystyle 3x\left(2x^{2}\right) +3x \left(-5x\right) + 3x \left(4\right)
\displaystyle =\displaystyle 6 x^{3} - 15 x^{2} + 12x

Since there are no more like terms and the expression is already in standard form, the final answer is 6 x^{3} - 15 x^{2} + 12x.

Reflect and check

6 x^{3} - 15 x^{2} + 12x is considered a polynomial of degree 3, since 3 is the value of the highest exponent on a variable in the polynomial.

Example 2

Consider the polynomials 7 y + 2 and 4 y - 5.

a

Find the product of the two polynomials.

Worked Solution
Create a strategy

Multiply the polynomials using distribution.

Apply the idea

We can create an area model to multiply the two polynomials:

A rectangle divided into 2 rows of 2 rectangles. The top left rectangle is labeled 28 y squared, the bottom left rectangle is labeled negative 35 y, the top right rectangle is labeled 8 y and the bottom right rectangle is labeled negative 10. The side of the rectangle is labeled 4 y next to the top right rectangle, and negative 5 next to the bottom right. The top of the rectangle is labeled 7 y + 2.

Combining each of the terms, we get:

28y^{2} - 35y + 8y - 10 = 28y^{2} - 27y - 10

Reflect and check

We can also use the distributive property to get the product of 7y + 2 and 4 y - 5.

\displaystyle \left( 7 y + 2\right) \left( 4 y - 5\right)\displaystyle =\displaystyle 4y\left( 7 y + 2\right) - 5\left( 7 y + 2\right)Distributive property
\displaystyle =\displaystyle 4y\left(7y\right) + 4y\left(2\right) -5\left(7y\right) - 5\left(2\right)Distributive property
\displaystyle =\displaystyle 28 y^{2} + 8y - 35 y - 10Distributive property
\displaystyle =\displaystyle 28y^{2} - 27y - 10Combine like terms

Since the expression is already in standard form, the final answer is 28y^{2} - 27y - 10.

b

Explain why the product of two polynomials is also a polynomial.

Worked Solution
Apply the idea

A polynomial is a collection of terms in the form mx^{n} where m is a real number and n is a non-negative integer.

We know that product of two algebraic terms with non-negative integer exponents together results in an algebraic term with non-negative integer exponents. Since multiplying polynomials together results a sum of such products, by definition the result is a polynomial expression.

Reflect and check

We can use this explanation to think about what happens when we perform multiple operations on polynomials. What happens if we add two polynomials and multiply this result by another polynomial? What if we multiply three polynomials? Is our result still a polynomial?

Example 3

Inhar is designing a cubic storage container with odd-numbered side lengths. They decide to let 2x+1 yards represent the length of each side.

a

Confirm that the side length will always be odd.

Worked Solution
Create a strategy

Analyze the given side length.

Apply the idea

Since 2x+1 represents the side lengths of the container, we can note that twice any number is always an even product. If we add an odd number like 1 to that, the side length 2x+1 will always be odd.

b

Write an expression for the surface area of the storage container.

Worked Solution
Create a strategy

Draw and label a diagram of the storage container first, then use it to calculate the surface area.

Apply the idea
A cube. The front view of the cube is divided into 2 rows of 2 squares. The side next to the top left square is labeled 2 x and the side next to the bottom left square is labeled 1. The bottom of the bottom left square is labeled 2 x and the bottom of the bottom right square is labeled 1.

We can calculate the area of one face on the storage container, then multiply the polynomial expression by 6 faces on the cube-shaped container.

A cube. The front view of the cube is divided into 2 rows of 2 squares. The top left square is labeled 4 x squared, the bottom left square is 2 x, the top right square is 2 x and the bottom right is 1. The side next to the top left square is labeled 2 x and the side next to the bottom left square is labeled 1. The bottom of the bottom left square is labeled 2 x and the bottom of the bottom right square is labeled 1.

Area of one face: 4x^{2}+2x+2x+1=4x^{2}+4x+1 square yards

Surface area of container: 6\left(4x^{2}+4x+1\right)=24x^{2}+24x+6 square yards

c

Write an expression for the volume of the storage container.

Worked Solution
Create a strategy

Use the formula for the volume of a cube to calculate the volume of the storage container.

Apply the idea

Since the formula for the volume of a cube is V=l \cdot w \cdot h, we can calculate the volume of the storage container as shown:

\displaystyle V\displaystyle =\displaystyle \left(2x+1\right)\left(2x+1\right)\left(2x+1\right)Substitute expressions for l,w, and h
\displaystyle =\displaystyle \left(2x+1\right)\left[\left(4x^{2}+2x+2x+1\right)\right]Distributive property
\displaystyle =\displaystyle \left(2x+1\right)\left(4x^{2}+4x+1\right)Combine like terms
\displaystyle =\displaystyle \left(8x^{3}+8x^{2}+2x\right)+\left(4x^{2}+4x+1\right)Distributive property
\displaystyle =\displaystyle 8x^{3}+12x^{2}+6x+1 \text{ cubic yards}Combine like terms
Reflect and check

A labeled diagram of the storage container can help us conceptualize the problem.

A cube with labels: Height: quantity 2 x + 1 yards for the height, Length: quantity 2 x + 1 yards for the length and Width: quantity 2 x + 1 yards for the width.

Example 4

Consider the diagram of the product of the expression \left(x-1\right)\left(x-4\right).

A diagram visualizing (x-1)(x-4), but most tiles are blank.
a

Find the missing values on the diagram to complete the visual representation of multiplying \left(x-1\right)\left(x-4\right).

Worked Solution
Create a strategy

Use the algebra tiles on the left side and the algebra tiles on the top row to find the area of the tiles with missing values.

Apply the idea

The diagram that shows the visual representation of multiplying \left(x-1\right)\left(x-4\right) is given by:

A diagram showing the visual representation of multiplying (x-1) by (x-4).
b

Write the product of \left(x-1\right)\left(x-4\right).

Worked Solution
Create a strategy

Add all like terms of the algebra tiles from part (a).

Apply the idea
\displaystyle \left(x-1\right)\left(x-4\right)\displaystyle =\displaystyle x^{2}-x-x-x-x-x+1+1+1+1Add all the algebra tiles
\displaystyle =\displaystyle x^{2}-5x+4Evaluate
Reflect and check

You may encounter some products of polynomials that require combining exponents with a degree greater than 1. Recall the expanded form of the product property for exponents that says ax^{m} \cdot bx^{n} = abx^{m+n}.

So, we have:

\displaystyle 5x^3 \cdot 4x^2\displaystyle =\displaystyle 5 \cdot x \cdot x \cdot x \cdot 4 \cdot x \cdot x
\displaystyle =\displaystyle 5 \cdot 4 \cdot x \cdot x \cdot x \cdot x \cdot x
\displaystyle =\displaystyle 20x^5
Idea summary

Polynomials can be multiplied using the distributive property. Using an area model for multiplying polynomials helps keep track of terms.

Special products of binomials

For some products of binomials, we can look for patterns to help us simplify more efficiently.

Exploration

Consider the expansions of the following binomials of the form \left(a+b\right)\left(a+b\right)=\left(a+b\right)^{2}:

  • \left(x+3\right)\left(x+3\right) = x^{2} + 6x + 9

  • \left(x+5\right)\left(x+5\right)= x^{2} + 10x+ 25

  • \left(x+6\right)^{2}=x^{2} + 12x + 36

  • \left(2r+3s\right)^{2}=4r^{2} + 12rs + 9s^{2}

  • \left(5s-3\right)^{2}= 25s^{2} - 30s + 9

  1. What do you notice about the linear coefficient of the product?

  2. What do you notice about the constant of the product?

  3. Is there a general rule for this type of product?

Consider the expansions of the following binomials of the form \left(a+b\right)\left(a-b\right):

  • \left(x+3\right)\left(x-3\right)=x^{2}-9

  • \left(x-5\right)\left(x+5\right)=x^{2}-25

  • \left(x+6\right)\left(x-6\right)=x^{2}-36

  • \left(2r+3s\right)\left(2r-3s\right)=4r^{2}-9s^{2}

  • \left(5s-3\right)\left(5s+3\right)=25s^{2}-9

  1. What do you notice about the linear coefficient of the product?

  2. What do you notice about the constant of the product?

  3. Is there a general rule for this type of product?

Some products of binomials follow special patterns.

For example, consider the product of a binomial squared, \left(a+b\right)^2:

An area model showing (a+b) multiplied by (a+b). Ask your teacher for more information.

\,\\\,\\\,We can expand \left(a+b\right)^{2} to \left(a+b\right)\left(a+b\right) and represent them with an area model. Evaluating with this model and combining like terms, we get the product a^{2} + 2ab + b^{2}.

So, we have:

\left(a+b\right)^2=\left(a+b\right)\left(a+b\right)= a^{2}+2ab + b^{2}

Now consider the product of a sum and a difference, \left(a+b\right)\left(a-b\right):

An area model showing a+b multiplied by a-b. Ask your teacher for more information.

Notice that the term ab and -ab are opposites and combine to make zero. We call this a zero pair. So, \left(a+b\right)\left(a-b\right) = a^{2} - b^{2}.

A binomial of the form a^{2} - b^{2} is called a difference of two squares.

If we remember the patterns for these special products, we can multiply two polynomials without using the distributive property.

For binomials, we have the following special binomial products, which are called identities:

Square of a Sum

\left(a + b\right)^{2} = a^{2} + 2 a b + b^{2}

Square of a Difference

\left(a - b\right)^{2} = a^{2} - 2 a b + b^{2}

Product of a sum and difference

\left(a+b\right)\left(a-b\right) = a^{2} - b^{2}

Note: \left(a + b\right)^{2} \neq a^{2} + b^{2} and \left(a - b\right)^{2} \neq a^{2} - b^{2}.

Examples

Example 5

Multiply and simplify the following binomials.

a

\left(x - 4\right)^{2}

Worked Solution
Create a strategy

We check first whether \left(x - 4\right)^{2} is a special binomial product and identify its form.

Apply the idea

Since \left(x - 4\right)^{2} is a square of a binomial in the form \left(a - b\right)^{2}, we use the formula and simplify the expression:

\displaystyle \left(a - b\right)^{2}\displaystyle =\displaystyle a^{2} - 2 a b + b^{2}Identity for the square of a binomial
\displaystyle \left(x - 4\right)^{2}\displaystyle =\displaystyle x^{2} - 2\left(x\right)\left(4\right) + 4^{2}Substitute a=x and b=4
\displaystyle =\displaystyle x^{2} - 8 x + 16Evaluate the multiplication and exponent
Reflect and check
An area model showing how to find the product of x-4 and x-4.

\,\\\,We can also use an area model to find the product.

Combining like terms, we get:

\left(x-4\right)^{2} = x^{2} + 16

b

\left(x + 4\right)\left(x - 4\right)

Worked Solution
Create a strategy

We check first whether \left(x + 4\right)\left(x - 4\right) is a special binomial product and identify its form.

Apply the idea

Since \left(x + 4\right)\left(x - 4\right) is a product of a sum and difference, we use the formula and simplify the expression:

\displaystyle \left(a+b\right)\left(a-b\right)\displaystyle =\displaystyle a^{2} - b^{2}Identity for the product of a sum and difference
\displaystyle \left(x+4\right)\left(x-4\right)\displaystyle =\displaystyle x^{2} - 4^{2}Substitute a=x and b=4
\displaystyle =\displaystyle x^{2} - 16Evaluate the exponent
Reflect and check

Using an area model, we see \left(x+4\right)\left(x-4\right) = x^{2} - 16.

An area model showing x+4 multiplied by x-4. ASk your teacher for more information.
c

\left(2x + 5\right)\left(2x - 5\right)

Worked Solution
Create a strategy

We check first whether \left(2x + 5\right)\left(2x - 5\right) is a special binomial product and identify its form.

Apply the idea

Since \left(2x + 5\right)\left(2x - 5\right) is a product of a sum and difference, we use the formula and simplify the expression:

\displaystyle \left(a+b\right)\left(a-b\right)\displaystyle =\displaystyle a^{2} - b^{2}Identity for the product of a sum and difference
\displaystyle \left(2x+5\right)\left(2x-5\right)\displaystyle =\displaystyle \left(2x\right)^{2} - 5^{2}Substitute a=2x and b=5
\displaystyle =\displaystyle 4x^{2} - 25Evaluate the exponents
d

Multiply and simplify: 3\left(2x + 5y\right)^{2}

Worked Solution
Create a strategy

We check first whether 3\left(2x + 5y\right)^{2} involves a special binomial product and identify its form.

Apply the idea

Since \left(2x + 5y\right)^{2} is a square of a binomial in the form \left(a + b\right)^{2}, we use the formula and simplify the expression:

\displaystyle \left(a + b\right)^{2}\displaystyle =\displaystyle a^{2} + 2 a b + b^{2}Identity for the square of a binomial
\displaystyle 3 \left(2x + 5y\right)^{2}\displaystyle =\displaystyle 3 \left[\left(2x\right)^{2} + 2\left(2x\right)\left(5y\right) + \left(5y\right)^{2}\right]Substitute a=2x and b=5y and multiply by 3
\displaystyle =\displaystyle 3 \left[4x^{2} + 20xy + 25y^{2}\right]Evaluate the exponents and multiplication
\displaystyle =\displaystyle 12x^{2} + 60xy + 75y^{2}Distributive property
Idea summary

Recognizing the patterns in special binomial factors may be helpful in multiplication problems and upcoming lessons. Remember the patterns:

  • Square of a binomial (Sum): \left(a + b\right)^{2} = a^{2} + 2 a b + b^{2}

  • Square of a binomial (Difference): \left(a - b\right)^{2} = a^{2} - 2 a b + b^{2}

  • Product of a sum and difference: \left(a+b\right)\left(a-b\right) = a^{2} - b^{2}

Outcomes

A.EO.2

The student will perform operations on and factor polynomial expressions in one variable.

A.EO.2b

Determine the product of polynomial expressions in one variable, using a variety of strategies, including concrete objects and their related pictorial and symbolic models, the application of the distributive property, and the use of area models. The factors should be limited to five or fewer terms (e.g., (4x + 2)(3x + 5) represents four terms and (x + 1)(2x^2 + x + 3) represents five terms).

What is Mathspace

About Mathspace