Learning Objectives

By the end of this section, you will be able to:

- Multiply a polynomial by a monomial
- Multiply a binomial by a binomial
- Multiply a trinomial by a binomial

Note

Before you get started, take this readiness quiz.

- Distribute: 2(x+3).

If you missed this problem, review Exercise 1.10.31. - Combine like terms: (x^{2}+9x+7x+63).

If you missed this problem, review Exercise 1.3.37.

## Multiply a Polynomial by a Monomial

We have used the Distributive Property to simplify expressions like 2(x−3). You multiplied both terms in the parentheses, x and 3, by 2, to get 2x−6. With this chapter’s new vocabulary, you can say you were multiplying a binomial, x−3, by a monomial, 2.

Multiplying a **binomial** by a **monomial** is nothing new for you! Here’s an example:

Exercise (PageIndex{1})

Multiply: 4(x+3).

**Answer**Distribute. Simplify.

Exercise (PageIndex{2})

Multiply: 5(x+7).

**Answer**5x+35

Exercise (PageIndex{3})

Multiply: 3(y+13).

**Answer**3y+39

Exercise (PageIndex{4})

Multiply: y(y−2).

**Answer**Distribute. Simplify.

Exercise (PageIndex{5})

Multiply: x(x−7).

**Answer**(x^{2}-7 x)

Exercise (PageIndex{6})

Multiply: d(d−11).

**Answer**(d^{2}-11d)

Exercise (PageIndex{7})

Multiply:(7x(2 x+y))

**Answer**Distribute. Simplify.

Exercise (PageIndex{8})

Multiply: 5(x(x+4 y))

**Answer**(5 x^{2}+20 x y)

Exercise (PageIndex{9})

Multiply: 2(p(6 p+r))

**Answer**(12 p^{2}+2 p r)

Exercise (PageIndex{10})

Multiply: (-2 yleft(4 y^{2}+3 y-5 ight))

**Answer**Distribute. Simplify.

Exercise (PageIndex{11})

Multiply: (-3 yleft(5 y^{2}+8 y-7 ight))

**Answer**(-15 y^{3}-24 y^{2}+21 y)

Exercise (PageIndex{12})

Multiply: 4(x^{2}left(2 x^{2}-3 x+5 ight))

**Answer**(8 x^{4}-24 x^{3}+20 x^{2})

Exercise (PageIndex{13})

Multiply: 2(x^{3}left(x^{2}-8 x+1 ight))

**Answer**Distribute. Simplify.

Exercise (PageIndex{14})

Multiply: 4(xleft(3 x^{2}-5 x+3 ight))

**Answer**(12 x^{3}-20 x^{2}+12 x)

Exercise (PageIndex{15})

Multiply: (-6 a^{3}left(3 a^{2}-2 a+6 ight))

**Answer**(-18 a^{5}+12 a^{4}-36 a^{3})

Exercise (PageIndex{16})

Multiply:((x+3) p)

**Answer**The monomial is the second factor. Distribute. Simplify.

Exercise (PageIndex{17})

Multiply:((x+8) p)

**Answer**(x p+8 p)

Exercise (PageIndex{18})

Multiply:((a+4) p)

**Answer**(a p+4 p)

## Multiply a Binomial by a Binomial

Just like there are different ways to represent multiplication of numbers, there are several methods that can be used to multiply a **binomial** times a binomial. We will start by using the Distributive Property.

### Multiply a Binomial by a Binomial Using the Distributive Property

Look at Exercise (PageIndex{16}), where we multiplied a binomial by a **monomial**.

We distributed the p to get: | |

What if we have (x + 7) instead of p? | |

Distribute (x + 7). | |

Distribute again. | |

Combine like terms. |

Notice that before combining like terms, you had four terms. You multiplied the two terms of the first binomial by the two terms of the second binomial—four multiplications.

Exercise (PageIndex{19})

Multiply:((y+5)(y+8))

**Answer**Distribute ( *y*+ 8).Distribute again Combine like terms.

Exercise (PageIndex{20})

Multiply:((x+8)(x+9))

**Answer**(x^{2}+17 x+72)

Exercise (PageIndex{21})

Multiply:((5 x+9)(4 x+3))

**Answer**(20 x^{2}+51 x+27)

Exercise (PageIndex{22})

Multiply:((2 y+5)(3 y+4))

**Answer**Distribute (3 *y*+ 4).Distribute again Combine like terms.

Exercise (PageIndex{23})

Multiply:((3 b+5)(4 b+6))

**Answer**(12 b^{2}+38 b+30)

Exercise (PageIndex{24})

Multiply:((a+10)(a+7))

**Answer**(a^{2}+17 a+70)

Exercise (PageIndex{25})

Multiply:((4 y+3)(2 y-5))

**Answer**Distribute. Distribute again. Combine like terms.

Exercise (PageIndex{26})

Multiply:((5 y+2)(6 y-3))

**Answer**(30 y^{2}-3 y-6)

Exercise (PageIndex{27})

Multiply:((3 c+4)(5 c-2))

**Answer**(15 c^{2}+14 c-8)

Exercise (PageIndex{28})

Multiply:((x-2)(x-y))

**Answer**Distribute. Distribute again. There are no like terms to combine.

Exercise (PageIndex{29})

Multiply:((a+7)(a-b))

**Answer**(a^{2}-a b+7 a-7 b)

Exercise (PageIndex{30})

Multiply:((x+5)(x-y))

**Answer**(x^{2}-x y+5 x-5 y)

### Multiply a Binomial by a Binomial Using the FOIL Method

Remember that when you multiply a binomial by a binomial you get four terms. Sometimes you can combine like terms to get a **trinomial**, but sometimes, like in Exercise (PageIndex{28}), there are no like terms to combine.

Let’s look at the last example again and pay particular attention to how we got the four terms.

[egin{array}{c}{(x-2)(x-y)} {x^{2}-x y-2 x+2 y}end{array}]

Where did the first term, (x^{2}), come from?

We abbreviate “First, Outer, Inner, Last” as FOIL. The letters stand for ‘**F**irst, **O**uter, **I**nner, **L**ast’. The word FOIL is easy to remember and ensures we find all four products.

[egin{array}{c}{(x-2)(x-y)} {x^{2}-x y-2 x+2 y} {F qquad Oqquad Iqquad L}end{array}]

Let’s look at (x+3)(x+7).

Distributive Property | FOIL |

Notice how the terms in third line fit the FOIL pattern.

Now we will do an example where we use the FOIL pattern to multiply two binomials.

Exercise (PageIndex{31}): How to Multiply a Binomial by a Binomial using the FOIL Method

Multiply using the FOIL method: ((x+5)(x+9))

**Answer**

Exercise (PageIndex{32})

Multiply using the FOIL method: ((x+6)(x+8))

**Answer**(x^{2}+14 x+48)

Exercise (PageIndex{33})

Multiply using the FOIL method: ((y+17)(y+3))

**Answer**(y^{2}+20 y+51)

We summarize the steps of the FOIL method below. The FOIL method only applies to multiplying binomials, not other polynomials!

MULTIPLY TWO BINOMIALS USING THE FOIL METHOD

When you multiply by the FOIL method, drawing the lines will help your brain focus on the pattern and make it easier to apply.

Exercise (PageIndex{34})

Multiply: (y−7)(y+4).

**Answer**

Exercise (PageIndex{35})

Multiply: (x−7)(x+5).

**Answer**(x^{2}-2 x-35)

Exercise (PageIndex{36})

Multiply: (b−3)(b+6).

**Answer**(b^{2}+3 b-18)

Exercise (PageIndex{37})

Multiply: (4x+3)(2x−5).

**Answer**

Exercise (PageIndex{38})

Multiply: (3x+7)(5x−2).

**Answer**(15 x^{2}+29 x-14)

Exercise (PageIndex{39})

Multiply: (4y+5)(4y−10).

**Answer**(16 y^{2}-20 y-50)

The final products in the last four examples were trinomials because we could combine the two middle terms. This is not always the case.

Exercise (PageIndex{41})

Multiply: (10c−d)(c−6).

**Answer**(10 c^{2}-60 c-c d+6 d)

Exercise (PageIndex{42})

Multiply: (7x−y)(2x−5).

**Answer**(14 x^{2}-35 x-2 x y+10 y)

Be careful of the exponents in the next example.

Exercise (PageIndex{44})

Multiply: (left(x^{2}+6 ight)(x-8))

**Answer**(x^{3}-8 x^{2}+6 x-48)

Exercise (PageIndex{45})

Multiply: (left(y^{2}+7 ight)(y-9))

**Answer**(y^{3}-9 y^{2}+7 y-63)

Exercise (PageIndex{47})

Multiply:((2 a b+5)(4 a b-4))

**Answer**(8 a^{2} b^{2}+12 a b-20)

Exercise (PageIndex{48})

Multiply:((2 x y+3)(4 x y-5))

**Answer**(8 x^{2} y^{2}+2 x y-15)

### Multiply a Binomial by a Binomial Using the Vertical Method

The FOIL method is usually the quickest method for multiplying two binomials, but it *only* works for binomials. You can use the Distributive Property to find the product of any two polynomials. Another method that works for all polynomials is the Vertical Method. It is very much like the method you use to multiply whole numbers. Look carefully at this example of multiplying two-digit numbers.

Now we’ll apply this same method to multiply two binomials.

Exercise (PageIndex{49})

Multiply using the Vertical Method: ((3 y-1)(2 y-6))

**Answer**It does not matter which binomial goes on the top.

(egin{array}{lll}{ ext { Multiply } 3 y-1 ext { by }-6 ext { . }}&& { ext { Multiply } 3 y-1 ext { by } 2 y ext { . }}& & &{qquadspace3 y-1} & & {dfrac{ spacespace imes 2 y-6}{quad-18 y+6}} & ext{partial product} & &

ParseError: EOF expected (click for details)

Callstack:at (Bookshelves/Algebra/Book:_Elementary_Algebra_(OpenStax)/06:_Polynomials/6.03:_Multiply_Polynomials), /content/body/div[4]/div[3]/div[1]/dl/dd/p[2]/span, line 1, column 3at wiki.page()at (Courses/Remixer_University/Username:_pseeburger/MTH_098_Elementary_Algebra/6:_Polynomials/6.3:_Multiply_Polynomials), /content/body/div[1]/pre, line 2, column 14

& ext{partial product} & ext{Add like terms.} && ext{product} end{array})Notice the partial products are the same as the terms in the FOIL method.

Exercise (PageIndex{50})

Multiply using the Vertical Method: ((5 m-7)(3 m-6))

**Answer**(15 m^{2}-51 m+42)

Exercise (PageIndex{51})

Multiply using the Vertical Method: ((6 b-5)(7 b-3))

**Answer**(42 b^{2}-53 b+15)

We have now used three methods for multiplying binomials. Be sure to practice each method, and try to decide which one you prefer. The methods are listed here all together, to help you remember them.

MULTIPLYING TWO BINOMIALS

To multiply binomials, use the:

- Distributive Property
- FOIL Method
- Vertical Method

Remember, FOIL only works when multiplying two binomials.

## Multiply a Trinomial by a Binomial

We have multiplied monomials by monomials, monomials by polynomials, and binomials by binomials. Now we’re ready to multiply a **trinomial** by a **binomial**. Remember, FOIL will not work in this case, but we can use either the Distributive Property or the Vertical Method. We first look at an example using the Distributive Property.

Exercise (PageIndex{52})

Multiply using the Distributive Property: ((b+3)left(2 b^{2}-5 b+8 ight))

**Answer**Distribute. Multiply. Combine like terms.

Exercise (PageIndex{53})

Multiply using the Distributive Property: ((y-3)left(y^{2}-5 y+2 ight))

**Answer**(y^{3}-8 y^{2}+17 y-6)

Exercise (PageIndex{54})

Multiply using the Distributive Property: ((x+4)left(2 x^{2}-3 x+5 ight))

**Answer**(2 x^{3}+5 x^{2}-7 x+20)

Now let’s do this same multiplication using the Vertical Method.

Exercise (PageIndex{55})

Multiply using the Vertical Method: ((b+3)left(2 b^{2}-5 b+8 ight))

**Answer**It is easier to put the polynomial with fewer terms on the bottom because we get fewer partial products this way.

Multiply (2 *b*^{2}− 5*b*+ 8) by 3.Multiply (2 *b*^{2}− 5*b*+ 8) by b.Add like terms.

Exercise (PageIndex{56})

Multiply using the Vertical Method: ((y-3)left(y^{2}-5 y+2 ight))

**Answer**(y^{3}-8 y^{2}+17 y-6)

Exercise (PageIndex{57})

Multiply using the Vertical Method: ((x+4)left(2 x^{2}-3 x+5 ight))

**Answer**(2 x^{3}+5 x^{2}-7 x+20)

We have now seen two methods you can use to multiply a **trinomial** by a binomial. After you practice each method, you’ll probably find you prefer one way over the other. We list both methods are listed here, for easy reference.

MULTIPLYING A TRINOMIAL BY A BINOMIAL

To multiply a trinomial by a binomial, use the:

- Distributive Property
- Vertical Method

Note

Access these online resources for additional instruction and practice with multiplying polynomials:

__Multiplying Exponents 1____Multiplying Exponents 2____Multiplying Exponents 3__

## Key Concepts

**FOIL Method for Multiplying Two Binomials**—To multiply two binomials:- Multiply the
**First**terms. - Multiply the
**Outer**terms. - Multiply the
**Inner**terms. - Multiply the
**Last**terms.

- Multiply the
**Multiplying Two Binomials**—To multiply binomials, use the:- Distributive Property (
__Example__) - FOIL Method (
__Example__) - Vertical Method (
__Example__)

- Distributive Property (
**Multiplying a Trinomial by a Binomial**—To multiply a trinomial by a binomial, use the:- Distributive Property (
__Example__) - Vertical Method (
__Example__)

- Distributive Property (

To multiply polynomials , you need to know how

1) to use the distributive law: ( quad a(b+c) = ab + ac quad ) or ( quad (b+c) a = b a + c a quad ), which is one of the basic rules of algebra ,

2) mutliply monomials,

3) and add like terms of a plynomial,

( ) ( ) ( ) ( ) ( ) Example 1

Multiply the following monomials and polynomials

a) ( 2 (6 x + 2) quad ) b) ( quad - 3 x (2 x^2 - x) quad )

c) ( quad -dfrac<1> <2>x^2 ( 4 x^2 - 2x + 6 x y) )

__Solution to Example 1__

a)

Given ( qquad 2 (6 x + 2) )

Mulitply constants together and variables together

( qquad qquad = 2(6)(x) + 2(2) )

Simplify

( qquad qquad = 12 x + 4 )

b)

Given ( qquad - 3 x (2 x^2 - x) )

Mulitply constants together and variables together

( qquad qquad = -3(2)(x x^2) -3(-1)(x x) )

Simplify

( qquad qquad = -6x^3 + 3x^2 )

c)

Given ( qquad -dfrac<1> <2>x^2 ( 4 x^2 - 2x + 6 x y) )

Mulitply constants together and variables together

( qquad qquad = -dfrac<1> <2>(4) (x^2 x^2) -dfrac<1> <2>(-2)(x^2 x) -dfrac<1> <2>(6) (x^2 x y) )

Simplify

( qquad qquad = - 2x^4 + x^3 - 3x^3 y )

## 6.3: Multiply Polynomials - Mathematics

General form of a polynomial in x:

a n x n + a n-1 x n-1 + a n-2 x n-2 + . . . + a 2 x 2 + a 1 x 1 + a 0 , where

Degree of a term is the sum of the exponents on the variables in the term.

The term 4x 3 y 5 has degree 8 since .

Degree of a polynomial is the degree of the highest degree term.

To write a polynomial in descending order for a certain variable means to write the polynomial from the term with the highest exponent (in the certain variable) on the left descending to the term with the lowest exponent (in the certain variable) on the right.

Examples of Polynomials in x | |||
---|---|---|---|

Name | Example | Degree | Note |

Monomial | 3x 2 | 2 | One term (mono) |

Binomial | 1 | Two terms (bi) | |

Trinomial | 3 | Three terms (tri) | |

Polynomial | 4 | Many terms (poly) |

Polynomials can be in more than one variable.

Examples of Polynomials in x and y | |||
---|---|---|---|

Name | Example | Degree | Note |

Monomial | 3x 2 y 3 | 5 | One term (mono) |

Binomial | 2 | Two terms (bi) | |

Trinomial | 7 | Three terms (tri) | |

Polynomial | 5 | Many terms (poly) |

Example: add and | |

remove parentheses | |

add like terms |

1) Remove parentheses (distribute "-" through). 2) Combine like terms.

Example: subtract from | |

remove parentheses | |

add like terms |

- FOIL only works when multiplying binomials--the distributive property works when multiplying any polynomials together.

Example: (2x + 3)(4x + 5) | |

(2x + 3)(4x + 5) = | |

= 2x(4x + 5) + | distributive property |

= (2x)(4x) + (2x)(5) + 3(4x) + 3(5) | distributive property again |

= 8x 2 + 10x + 12x + 15 | simplifying |

= 8x 2 + 22x + 15 | combine like terms |

Example: (2x - 3)(4x 2 - 5x + 6) | |

(2x - 3)(4x 2 - 5x + 6) = | |

= 2x(4x 2 - 5x + 6) - | distributive property |

= (2x)(4x 2 ) - (2x)(5x) + (2x)(6) - 3(4x 2 ) + 3(5x) - 3(6) | distributive property again |

= 8x 3 - 10x 2 + 12x - 12x 2 + 15x - 18 | simplifying |

= 8x 3 - 22x 2 + 27x - 18 | combine like terms |

(a + b) 2 = a 2 + 2ab + b 2 (a - b) 2 = a 2 - 2ab + b 2 (a + b)(a - b) = a 2 - b 2 Note: the a and b may be any algebraic expression.

## Multiplying Trinomials and Polynomials

1)

First, we distribute the and get

Next, we distribute the 3 and get

Now we have , but we are not finished because there is a set of like terms that we can add together. Add 4x 2 and 3x 2 to get 7x 2 . Also add 1x and 12x to get 13x. Make sure the final answer is in standard form:.

2)

First, we distribute the and get

Next, we distribute theand get

Then, we distribute theand get

Now we have , but we are not finished because we can combine like terms. Combine them to get our final answer (in standard form): .

As you can see, it doesn't matter how many terms are in each polynomial. You just keep distributing. It's not hard, but you do have to be very careful in your work.

**Practice:** : Multiply the polynomials

1)

2)

3)

4)

5)

## Algebra: Multiplying Polynomials

Unlike addition and subtraction, you don't need like terms in order to multiply polynomials (nor do you need like terms to divide polynomials, but I'll discuss that in the next section). In fact, multiplying polynomials is actually pretty easy. All you have to do is apply exponential rules and the distributive property, both of which you learned in Encountering Expressions.

### Products of Monomials

Here's what you should do to multiply two monomials together:

**Multiply their coefficients**. The result is the coefficient of the answer.**List all the variables that appear in either term**. These should follow the coefficient you got in step 1, preferably in alphabetical order.**Add up the powers**. Determine the sums of matching variables' exponents and write them above the corresponding variable in the answer.

##### How'd You Do That?

Step 3 tells you to add the powers of matching variables because of the exponential rule from Encountering Expressions stipulating that *x* *a* *x* *b* = x *a + b* . (The product of exponential expressions with matching bases equals the base raised to the sum of the powers.)

Even if the steps seem weird at first, don't worry. Multiplying monomials is a skill you'll understand very quickly.

**Example 3**: Calculate the products.

- (a) (-3
*x*2*y*3*z*5 )(7*xz*3 ) **Solution**: First multiply the coefficients: -3 7 = -21 then, list all the variables that appear in the problem in alphabetical order. (It doesn't matter that the second monomial doesn't contain a*y*. As long as a variable appears anywhere in the problem, you should list it next to the coefficient you just found.)- -21
*xyz* - Add up the exponents for each variable you listed. The first term has
*x*to the 2 power, and the second term has*x*to the 1 power, so the answer will have*x*to the 2 + 1 = 3 power. Similarly, the*z*power of the answer should be 8, since there's a z to the 5 power in the first monomial and a z to the 3 in the second. Since there's only one y term, you just copy its power to the final answer there's nothing to add. - -21
*x*3*y*3*z*8 - (b) 3
*w*2*x*(2*wxy*-*x*2*y*2 ) **Solution**: Apply the distributive property, multiplying both terms by 3*w*2*x*.- 3
*w*2*x*(2*wxy*) + 3*w*2*x*(-*x*2*y*2 ) - Find each product separately.
- 6
*w*3*x*2*y*- 3*w*2*x*3*y*2

##### You've Got Problems

Problem 3: Calculate the product.

3*x* 2 *y* (5*x* 3 + 4*x* 2 *y* - 2*y* 5 )

### Binomials, Trinomials, and Beyond

##### Critical Point

Some algebra teachers focus on the FOIL method, a technique for multiplying two binomials. Each letter stands for a pair of terms in the binomials, the first, outside, inside, and last terms.

If you've never heard of FOIL, that's fine, because it only works for the special case of multiplying two binomials, whereas my multiple distribution technique works for all polynomial products. Besides, if you use my method, you actually end up doing FOIL anyway, even though it's unintentional.

##### Kelley's Cautions

Once you multiply, always make sure to see if you can simplify the result. Just about every algebra teacher in the world demands simplified answers, and if you don't comply, they've been known to do things like mark answers wrong, take points off, or (in extreme cases) get so angry that they send a cybernetic organism back in time to kill you before you sign up for their class.

Calculating polynomial products is kind of freeing. As I've said, two terms need not have anything in common to be multiplied together. (Based on couples I've met, I think the same is true for people, but I digress.) However, so far you can only multiply polynomial expressions if one of them is a monomial. In Example 3(a), you had two monomials, and in Example 3(b) and Problem 3, you were distributing a monomial. It turns out that multiplying polynomials with more than one term can be accomplished through a slightly modified version of the distributive property.

##### You've Got Problems

Problem 4: Find the product and simplify. (2*x* + *y*)(*x* - 3*y*)

Thanks to the distributive property, you already know that the expression *a*(*b* + *c*) can be rewritten as *ab* + *ac* just multiply the *a* by each thing in the parentheses. In a similar fashion, you can calculate the product of the expression (*a* + *b*)(*c* + *d*), even though in this case, you're multiplying binomials. Instead of just distributing *a*, like you did moments ago, you'll distribute each term in the first binomial through the second binomial, one at a time.

In other words, you'll multiply everything in the second binomial by *a* and then go through and do it again, this time multiplying everything by *b*.

So, you're still distributing, you're just doing it twice, that's all. What if you were multiplying a trinomial by a trinomial? Follow the same procedure distribute each term in the first polynomial through the second, one at a time.

- (
*a*+*b*+*c*)(*d*+*e*+*f*) =*ad*+*ae*+*af*+*bd*+*be*+*bf*+*cd*+*ce*+*cf*

In case you're wondering, the numbers of terms in the polynomials don't have to match. You could multiply a binomial times a trinomial just as easily, as you'll see in Example 4.

**Example 4**: Find the product and simplify.

**Solution**: Each term of the left polynomial, *x* and -2*y*, should be distributed through the second polynomial, one at a time.

- (
*x*)(*x*2 ) + (*x*)(2*xy*) + (*x*)(-*y*2 ) + (-2*y*)(*x*2 ) + (-2*y*)(2*xy*) + (-2*y*)(-*y*2 )

If you place all of the terms in parentheses, you don't have to worry about signs right away. It doesn't matter if some terms are positive and some are negative just write them all in parentheses and add all the products together.

Now all you have to do is multiply pairs of monomials together.

The directions for the problem tell you to simplify, which means you should now look for like terms which can be combined. If you look closely, you'll see that the terms 2*x* 2 *y* and -2*x* 2 *y* have the same variable, so they can be combined to get 0 (they're opposites of one another, so they'll cancel each other out). In addition, you can combine the terms -*xy* 2 and -4*xy* 2 to get -5*xy* 2 .

Excerpted from The Complete Idiot's Guide to Algebra 2004 by W. Michael Kelley. All rights reserved including the right of reproduction in whole or in part in any form. Used by arrangement with **Alpha Books**, a member of Penguin Group (USA) Inc.

## Symbolab Blog

Multiplying polynomials can be tricky because you have to pay attention to every term, not to mention it can be very messy. There are a few ways of multiplying polynomials, depending on how many terms are in each polynomial. In this post, we will focus on how to multiply two term polynomials and how to multiply two or more term polynomials.

__Multiply two term polynomials__

When multiplying polynomials with two terms, you use the FOIL method. The FOIL method only works for multiplying two term polynomials. FOIL stands for first, outer, inner, last. This lets you know the order of how to distribute and multiply the terms. Let’s see how it works.

Multiplying these polynomials is pretty simple because if you memorize these identities then you just plug in the values and have an answer.

__Multiplying multiple term polynomials__ You cannot use the FOIL method to multiply these polynomials. Instead, you have to multiply each term in one polynomial by each term in the other. You can do this by multiplying each term of one polynomial by the other polynomial. This can be tricky because it is easy to miss one term. When we do examples of this, it will become easier to understand how to solve them.

When multiplying polynomials, you may come across multiplying variables with exponents by variables with exponents. In this case, we use this exponent rule:

For this rule, the base or variable must be the same. When multiplying variables with exponents, you add the exponents together.

Let’s see some examples to understand how to multiply polynomials.

First example (click here):

## FAQs on Multiplying Polynomials

### How do you Multiply Three Polynomials?

Multiplication of three polynomials is a two-step process that involves the following two steps:

- Multiplication of coefficients
- Multiplication of the variables using Laws of Exponents as and when required.

Let's take an example to understand the multiplication of three polynomials. **Example:** Multiply (3m+2), 4n 2 , and 7p.

- The above given three polynomials are written as (3m+2)× 4n 2 × 7p
- By using distributive property of polynomial multiplication we get, ((3m× 4n 2 )+(2× 4n 2 ))× 7p = (12mn 2 + 8n 2 )7p = 84mn 2 p + 56n 2 p

Thus, the above multiplication can be shown as (3m+2)× 4n 2 × 7p = 84mn 2 p + 56n 2 p.

### How can we Multiply Polynomials Using the Box Method?

Two or more polynomials can be multiplied using the box method. The terms are written across a box and their corresponding products are written within the box. **Example:** (3x 2 +2x+4)(4x+5)

3x 2 +2x+4 will be written on the vertical side of the box while 4x+5 will be written on the horizontal side of the box, or vice-versa. Then, first, we will multiply 3x 2 by 4x, then 3x 2 by 5, and write the products in the corresponding section of the box. Secondly, we will multiply 2x by 4x and 2x by 5 and write down the products. The final column of the box is filled by multiplying 4 by 4x and 4 by 5. At last, we will add all six terms obtained to get the final answer.

Therefore, the result of the multiplication of both the polynomials is (12x 3 +23x 2 +26x+20).

### How do you Multiply Binomials Using the Grid Method?

The steps to multiply polynomials by a box method or the grid method is as follows: **Example:** (x+6)(2x+3)

x+6 will be written on the vertical side of the box while 2x+3 will be written on the horizontal side of the box, or vice-versa. Multiply each term with the respective terms. Therefore, the product which we get is (2x 2 +15x+18).

### How Many Methods are there for Multiplying Polynomials?

There are two methods for multiplying polynomials:

### What does FOIL Stand for in Multiplying Binomials?

FOIL stands for First, Outer, Inner Last in multiplying binomials. The binomials are multiplied as:

- Step 1: Multiply the first term of each binomial.
- Step 2: Now multiply the outer term of each binomial.
- Step 3: Once this is done, now multiply the inner terms of the binomials.
- Step 4: Now the last terms are multiplied.
- Step 5: Once all the above four steps are done, the products obtained as each step are added, like terms are combined and the answer is simplified.

### What is the Best Method for Multiplying Polynomials?

The best method for multiplying polynomials is the distributive property of multiplying polynomials. The steps to multiply a polynomial using the distributive property are:

- Step 1: Write both the polynomials together.
- Step 2: Out of the two brackets, keep one bracket constant.
- Step 3: Now multiply each and every term from the other bracket.

### How Do You Multiply Two Trinomials Together?

Two trinomials can be multiplied together by using the box method as well as distributive property. Let's take an example to understand the multiplication of two trinomials.

**Example:** Multiply (5xy+2x+3) with (x 2 +3xy+7)

- The above given two trinomials are written as (5xy+2x+3)× (x 2 +3xy+7)
- By using distributive property of polynomial multiplication we get, (5xy+2x+3)× (x 2 +3xy+7) = 5x 3 y + 15x 2 y 2 + 2x 3 + 6x 2 y + 44xy+ 3x 2 + 14x + 21

Thus, the above multiplication can be shown as (5xy+2x+3)× (x 2 +3xy+7) = 5x 3 y + 15x 2 y 2 + 2x 3 + 6x 2 y + 44xy + 3x 2 + 14x + 21.

## Questions

<a href=”/intermediatealgebraberg/back-matter/answer-key-6-5/”>Answer Key 6.5

## Related wikiHows

## PRODUCTS OF POLYNOMIALS

### OBJECTIVES

- Find the product of two binomials.
- Use the distributive property to multiply any two polynomials.

In the previous section you learned that the product A(2x + y) expands to A(2x) + A(y).

Now consider the product (3x + z)(2x + y).

Since (3x + z) is in parentheses, we can treat it as a single factor and expand (3x + z)(2x + y) in the same manner as A(2x + y). This gives us

If we now expand each of these terms, we have

Notice that in the final answer each term of one parentheses is multiplied by every term of the other parentheses.

Note that this is an application of the distributive property. |

Note that this is an application of the distributive property. |

Since - 8x and 15x are similar terms, we may combine them to obtain 7x.

In this example we were able to combine two of the terms to simplify the final answer.

Here again we combined some terms to simplify the final answer. Note that the order of terms in the final answer does not affect the correctness of the solution.