Source: Melanie Shebel

What is a Polynomial?

The best way to tackle any problem, whether it’s in math, chemistry, or even life, is to understand the nature of the beast. Let’s go over some quick rules of polynomials.

A polynomial can be made up of variables (like x or y), constants (these are numbers like 3, 5, and 11), and exponents (such as the 2 in in x2.) A number attached to a variable is called a coefficient.

In 2x + 4, four is the constant and two is the coefficient of x.

Polynomials can 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. The above example has three terms: 3y2, 2x, and 5

(3)(2x) is like saying 3 times 2 times x.

Multiply three times two times x to get 6x

Multiplying Monomials
Let’s start by multiplying monomials. These are polynomials with just one term (which make them a great place to begin.) When you’re multiplying polynomials, regardless of how many terms they contain, you’ll be taking it just two terms at a time.

We’ll begin with these two monomials:

What 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. (It’s a good idea to get into the habit of using · for multiplication. Using “x” to denote multiplication can get confusing since it looks like the variable ‘x.’)

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.

Quick Refresher in 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 Binomials by Monomials
Now that you’ve got multiplying monomials down pat, let’s move on to multiplying a monomial and a binomial (a polynomial with two terms.)

When multiplying a binomial by a monomial, you have to distribute the monomial into the parenthesis.

Write down 3x times 4x + 3x times 2x.

3x times 4x is 12x² and 3x times 2y is 6xy.

Sample problem:

  1. Multiply 3x times 4x. Write down the product.
  2. Write down a plus sign, since there’s addition in the parenthesis and the product of 3x and 2y is positive.
  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.

The FOIL method

Multiply the first terms, the outer, inner, and then finally the last terms. Combine like terms and voilà, you’ve got FOIL down pat!

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.

Sample problem:
(x+2) (x+1)

  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.)
  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.)
  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.)
  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
  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

Distribute each term in one polynomial to each term in the other polynomial.

The main drawback to FOIL is that it only works if you’re multiplying two binomials. However, you can still distribute terms in a way similar to foil, but it can get a little messier (and thus trickier.)

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)

  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.

  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.
  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.
  4. Combine like terms.
    x4-11x3+6x2+5x2-55x+30  =  x4-11x3+11x2+-55x+30


The Grid Method

Start off with a grid containing the terms of one polynomial across the top and the terms of the other down the side.

As mentioned earlier, FOIL can only be used when multiplying two binomials and the distribution method can easily become a tangled mess, especially with larger polynomials. One of the best ways to multiply polynomials is the grid method.

The grid method is really identical to the distribution method except everything goes right into a handy grid. This helps you keep track of everything, 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 something scary with 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!

Multiply term in the first row by the term in the first column. Write down the product.

Continue by filling in the next box with the product of the terms in the corresponding column and row

Fill in each box in the grid.

Here we’re starting on the next row.

Continue finding the products of terms

Yay! We’ve got all the products we need! The hard part is done!

Group together like terms (this will make it easier to find all the sums and differences.)

Combine the like terms.

Yay! You’re done!