How to Multiply Polynomials, with Examples
What is a Polynomial?
A polynomial can be made up of variables (such as x and y), constants (such as 3, 5, and 11), and exponents (such as the 2 in x2.) Remember that constants must be on their own. A number attached to a variable is called a coefficient.
In 2x + 4, 4 is the constant and 2 is the coefficient of x.
Polynomials must contain addition, subtraction, or multiplication, but not division. They also cannot contain negative exponents.
The following example is a polynomial containing variables, constants, addition, multiplication, and a positive exponent:
3y2 + 2x + 5
Each segment in a polynomial separated by addition or subtraction is called a term (also known as a monomial.)
Multiplying a Monomial Times a Monomial
Before jumping into multiplying polynomials, let's break it down into multiplying monomials. When you're multiplying polynomials, you'll be taking it just two terms at a time, so getting monomials down is important.
Let's start with:
All you need to do here is break it down to 3 times 2 times x. You can get rid of the parenthesis and write it out like 3 · 2 · x. (Avoid using "x" to mean multiplication. It can get confusing with the letter x as a variable. Use · for multiplication instead!)
Because of the commutative property of multiplication, you can multiply the terms in any order, so let's solve this by going from left to right:
3 · 2 · x
3 times 2 is 6, so we're left with:
6 · x, which can be written as 6x.
Practice What You've Learned: Multiplying Monomials
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Quick Refresher on Multiplying Exponents
When adding exponents, you add the coefficients.
2x + 3x = 5x.
x + x = 2x
So what do you do when multiplying exponents?
x · x = ?
When multiplying like variables with exponents, you just add the exponents.
(x2)(x3) = x5
This is the same as saying x · x · x · x · x
(2x)(5xy) = 10x2y
This is the same as saying 2 · x · 5 · x · y or 2 · 5 · x · x · y
Remember that x = x1. If no exponent is written, it's assumed that it's to the first power. This is because any number is equal to itself to the first power.
Multiplying 1 Term by 2 TermsClick thumbnail to view full-size
Multiplying 1 Term by 2 Terms
When multiplying one term by two terms, you have to distribute them into the parenthesis.
Step 1: Multiply 3x times 4x. Write down the product.
Step 2: Write down a plus sign, since there's addition in the parenthesis and the product of 3x and 2y is positive.
Step 3: Multiply 3x times 2y. Write down the product.
You should have 12x2 + 6xy written down. Since there are no like terms to add together, you're done.
If you're dealing with negative numbers or subtraction, you have to watch the signs.
For example, if the problem is -3x(4x+2y), you'll have to multiply negative 3x times everything in the parenthesis. Since the product of -3x and 4x is negative, you would have -12x2. Then, it would be -6xy since the product of -3x and 2y are negative (if the plus sign throws you off, you can write it as 12x2 + -6xy.
The FOIL method
Watch your signs:
The product of a positive multiplied by a positive will be positive.
The product of a negative multiplied by a negative will be positive.
The product of a positive multiplied by a negative will be negative.
Multiplying Binomials using the FOIL Method
A polynomial with just two terms is called a binomial. When you're multiplying two binomials together, you can use an easy to remember method called FOIL. FOIL stands for First, Outer, Inner, Last.
Step 1: Multiply the first terms in each binomial. The first terms here are the x from (x+2) and the x from (x+1). Write down the product. (The product of x times x is x2.)
Step 2: Multiply the outer terms in each of the two binomials. The outer terms here are the x from (x+2) and the 1 from (x+1). Write down the product. (The product of x times 1 is 1x, or x.)
Step 3: Multiply the inner terms in the two binomials. The inner terms here are the 2 from (x+2) and the x from (x+1). Write down the product. (The product of 2 times x is 2x.)
Step 4: Multiply the last terms in each of the two binomials. The last terms here are the 2 from (x+2) and the 1 from (x+1). Write down the product. (The product of 1 times 2 is 2.)
You should have: x2 + x + 2x + 2
Step 5: Combine like terms. There is nothing here with an x2 attached to it, so x2 stays as is, x and 2x can be combined to equal 3x, and 2 stays as is because there are no other constants.
Your final answer is: x2 + 3x + 2
Distributing Terms Without FOIL
Practice What You've Learned: Multiplying Polynomials
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Distributing Polynomials (Without FOIL)
When you're dealing with the multiplication of two polynomials, order them so that the polynomial with fewer terms is to the left. If the polynomials have an equal number of terms, you can leave it as is.
For example, if your problem is: (x2-11x+6)(x2+5)
Rearrange it so it looks like: (x2+5)(x2-11x+6)
Step 1: Multiply the first term in the polynomial on the left by each term in the polynomial on the right. For the problem above, you would multiply x2 by each x2,-11x, and 6.
You should have x4-11x3+6x2.
Step 2: Multiply the next term in the polynomial on the left by each term in the polynomial on the right. For the problem above, you would multiply 5 by each x2,-11x, and 6.
Now, you should have x4-11x3+6x2+5x2-55x+30.
Step 3: Multiply the next term in the polynomial on the left by each term in the polynomial on the right. Since there are no more terms in the left polynomial in our example, you can go ahead and skip to step 4.
Step 4: Combine like terms.
x4-11x3+6x2+5x2-55x+30 = x4-11x3+11x2+-55x+30
Multiplying Using a Grid
Using the Grid Method
One of the biggest drawbacks of using the FOIL method is that it can only be used for multiplying two binomials. Using the distribution method can get really messy, so it's easy to forget to multiply some terms.
One of the best ways to multiply polynomials is the grid method. This is actually just like the distribution method except everything goes right into a handy grid making it almost impossible to lose terms. Another thing that's nice about the grid method is that you can use it to multiply any type of polynomials whether they're binomials or have twenty terms!
Start off by making a grid. Put each term in one of the polynomials across the top and the terms of the other polynomial down the left side. In each box in the grid, fill in the product of the term for the row times the term for the column. Combine like terms and you're done!
Leave a comment below if you're still struggling. I want to create the perfect guide to multiplying polynomials and if there is something you don't quite understand or if there are any tricks that work for you, I want to know!