Multiplying Polynomials

 

Learning Objective(s)

·         Multiply polynomials and collect the like terms of the resulting sum of monomials.

 

Introduction

 

Multiplying polynomials involves applying the rules of exponents and the distributive property to simplify the product. This multiplication can also be illustrated with an area model and can be useful in modeling real world situations. Understanding polynomial products is an important step in factoring and solving algebraic equations.

 

The Product of a Monomial and a Polynomial

 

The distributive property can be used to multiply a polynomial by a monomial. Just remember that the monomial must be multiplied by each term in the polynomial. Consider the expression 2x(2x2 + 5x + 10).

 

This expression can be modeled with a sketch like the one below. This model is called an area model because the rectangular pieces represent the area created by the multiplication of the monomial and the polynomial.

 

 

2x2

5x

10

 

2x

 

4x3

 

10x2

 

20x

 

 

 

 

 

 

 

We can see that the product of the width, 2x, and the length, 2x2 + 5x + 10, is the area of the entire shaded region. The area can be split into three smaller pieces. Each of those pieces has a width of 2x and a length represented by one of the terms of the polynomial.

 

Area models are a helpful way to visualize a multiplication problem. But we can also find the product of two polynomials algebraically, by applying the distributive property. Remember that the distributive property says that multiplying a sum by a number is the same as multiplying each addend by the number and then adding: a(b + c) = ab + ac. It doesn't matter how many terms there are: a(b + c + d) = ab + ac + ad.

 

Let's try one:

 

Example

Problem

 

5x3(4x2 + 3x + 7)

 

 

 

 

 

Distribute the monomial to each term of the polynomial

 

 

Add the products

Answer

 

 

 

 

 

Product of Two Binomials

 

Now let's explore multiplying two binomials. Once again, we can draw an area model to help us make sense of the process. We'll use each binomial as one of the dimensions of a rectangle, and their product as the area.

 

The model below shows (x + 4)(2x + 2):

 

 

x

1

1

1

1

 

 

x

 

 

 

 

x2

 

 

x

 

 

x

 

 

x

 

 

x

 

 

x

 

 

 

 

x2

 

 

x

 

 

x

 

 

x

 

 

x

1

x

1

1

1

1

1

x

1

1

1

1

 

Each binomial is expanded into individual variables and numbers, x + 4 along the top of the model and 2x + 2 along the left side. The product of each pair of terms is a colored rectangle. The total area is the sum of all of these small rectangles, which is also the final product of multiplying the binomials. If we combine all the like terms, we can write the product, or area, as 2x2 + 10x + 8.

 

We can also use algebra to determine the product of two binomials. Just multiply each term in one binomial by all the terms in the other binomial as shown below:

 

Example

Problem

(x + 4)(2x + 2)

 

 

 

 

 

x(2x + 2) + 4(2x + 2)

 

 

Multiply each term in one binomial by each term in the other binomial

 

2x2 + 2x + 8x + 8

 

 

Rewrite to group like terms together

 

2x2 + 10x + 8

 

 

Combine like terms

Answer

2x2 + 10x + 8

 

 

 

 

Look back at the rectangle and see where each piece of 2x2 + 2x + 8x + 8 comes from. Can you see where we multiply x by 2x + 2, and where we get 2x2 from x(2x)?

 

Because multiplication is commutative, the terms can be multiplied in either order. The expression (2x + 2)(x  + 4) has the same product as (x  + 4)(2x + 2), both having a product of 2x2 + 10x + 8. (Work it out and see.) The order in which we multiply binomials does not matter. What matters is that we multiply each term in one binomial by each term in the other binomial.

 

The last step in multiplying polynomials is to combine like terms. Remember that a polynomial is simplified only when there are no like terms remaining.

 

Find the product:

 

(a + 10)(2a – 7)

 

A) 2a2 + 19a – 70

 

B) 3a + 3

 

C) 2a2 – 70

 

D) 2a2 + 13a – 70

 

Show/Hide Answer

A) Incorrect. The middle term should be 20a – 7a, which is 13a. The correct answer is  D) 2a2 + 13a – 70.

 

B) Incorrect. Multiply, don’t add, the terms in one binomial by the terms in the other binomial.  The correct answer is D) 2a2 + 13a – 70.

 

C) Incorrect. Multiply each term in one binomial by the terms in the other binomial. (a)(-7) and (10)(2a) are missing.  The correct answer is D) 2a2 + 13a – 70.

 

D) Correct.  a(2a – 7) + 10(2a – 7) = 2a2 – 7a + 20a – 70 = 2a2 + 13a – 70

 

 

 

Multiplication of binomials is sometimes needed to solve geometry problems. Suppose we want to find the area of a triangle with a base of 4x – 10 and a height of 2x + 3. To find the area of the triangle, we find the product of ½ of the base and the height (that's the area formula for triangles). This can be shown by the expression (4x – 10)(2x + 3).

 

Example

Problem

 (4x – 10)(2x + 3)

 

 

 [4x(2x + 3) – 10(2x + 3)]

 

Distribute multiplication

[8x2 + 12x – 20x – 30]

 

 

Multiply. Be careful about the negative sign when distributing –10 to 2x + 3.

 [8x2 – 8x – 30]

 

 

Combine like terms

––

 

Simplify

Answer

 

4x2 – 4x –15

 

 

 

 

Product of a Binomial and a Trinomial

 

Another type of polynomial multiplication problem is the product of a binomial and trinomial. Here again, the process is the same as with the other types of problems. Each term in the binomial must be multiplied by each of the terms in the trinomial. Two examples are shown below.

 

Example

Problem

 (3x + 6)(5x2 + 3x –10)

 

 

 

 3x(5x2 + 3x – 10) + 6(5x2 + 3x – 10)

 

Multiply each term in the binomial by the polynomial

 

(15x3 + 9x2 – 30x) + (30x2 + 18x – 60)

 

Use distributive property

 

15x3 + 9x2 – 30x + 30x2 + 18x – 60

 

Rewrite without parentheses

 

15x3 + 9x2 + 30x2 – 30x + 18x – 60

 

Regroup like terms to combine

 

15x3 + 39x2 – 12x – 60

 

Combine like terms

Answer

15x3 + 39x2 – 12x – 60

 

 

 

 

The next example shows multiplication by a binomial and polynomial that each contain subtraction.

 

Example

Problem

 (2p – 1)(3p2 – 3p ­+ 1)

 

 

 

 2p(3p2 – 3p ­+ 1) – 1(3p2 – 3p ­+ 1)

 

Multiply each term in the binomial by the polynomial

 

(6p3 – 6p2 ­+ 2p) – 1(3p2) – 1(-3p) – 1(1)

 

Be careful of the negative sign when distributing -1 to the trinomial.

 

6p3 – 6p2 ­+ 2p – 3p2 + 3p – 1

 

Rewrite without parentheses

 

6p3 – 6p2 ­– 3p2 + 2p + 3p – 1

 

Regroup like terms to combine

 

6p3 – 9p2 ­+ 5p – 1

 

Combine like terms

Answer

6p3 – 9p2 ­+ 5p – 1

 

 

 

 

Find the product:

 

(3x – 2)(2x2 + 4x – 11)

 

A) 6x3 + 8x2 – 41x + 22

 

B) 6x3 + 8x2 – 41x – 22

 

C) 6x3 + 12x + 22

 

D) 3x3 + 8x2 + 25x – 22

 

 

Show/Hide Answer

A) Correct. 3x(2x2 + 4x – 11) – 2(2x2 + 4x – 11) = 6x3  + 12x2 – 33x – 4x2 – 8x + 22 = 6x3  + 8x2 – 41x + 22

 

B) Incorrect. (-2)(-11) = 22. The correct answer is A, 6x3 + 8x2 – 41x + 22.

 

C) Incorrect. Each term in the binomial must be multiplied by each term in the trinomial. The correct answer is A, 6x3 + 8x2 – 41x + 22.

 

D) Incorrect. Multiply the coefficients (3x)(2x2); combine like terms by adding/subtracting coefficients according to their signs. The correct answer is A, 6x3 + 8x2 – 41x + 22.

 

 

Summary

 

Multiplication of binomials and polynomials requires use of the distributive property and integer operations. Whether the polynomials are monomials, binomials, or trinomials, carefully multiply each term in one polynomial by each term in the other polynomial. Be careful to watch the addition and subtraction signs and negative coefficients.  A product is written in simplified form if all of its like terms have been combined.