Multiply Polynomials (With Examples): FOIL & Grid Methods

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 x 2). Polynomials must contain addition, subtraction, or multiplication, but not division. They also cannot contain negative exponents. Each segment in a polynomial that is separated by addition or subtraction is called a term (also known as a monomial). The following example is a polynomial containing variables, constants, addition, multiplication, and a positive exponent: 3y 2 + 2x + 5.

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. Because of the commutative property of multiplication, you can multiply the terms in any order. In a practice problem like (3)(2x), all you need to do here is break it down to 3 times 2 times x. You can write it out like 3 · 2 · x, and since 3 times 2 is 6, we're left with 6x.

Quick Refresher on Multiplying Exponents

When multiplying like variables with exponents, you just add the exponents. (x 2)(x 3) = x 5. This is the same as saying x · x · x · x · x. Remember that x = x 1. 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 Terms

When multiplying one term by two terms, you have to distribute them into the parenthesis. Sample Problem: 3x(4x+2y). Multiply 3x times 4x, then multiply 3x times 2y. You should have 12x 2 + 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.

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. To solve a problem like (x+2) (x+1), follow these steps:

  1. Step 1: Multiply the first terms in each binomial. (The product of x times x is x 2.)
  2. Step 2: Multiply the outer terms in each of the two binomials. (The product of x times 1 is x.)
  3. Step 3: Multiply the inner terms in the two binomials. (The product of 2 times x is 2x.)
  4. Step 4: Multiply the last terms in each of the two binomials. (The product of 1 times 2 is 2.)
  5. Step 5: Combine like terms.

The FOIL method ensures you multiply the first terms, the outer, inner, and then finally the last terms to combine like terms and voila!

Mastering Multiplication Tables and Charts

The 10 times table is really very simple! You just need to understand a very simple rule: when you multiply a number by 10, just add a zero after the number to obtain the result of the multiplication. To help you understand, here are the results of the 10 times table:

  • 10 x 1 = 10
  • 10 x 2 = 20
  • 10 x 3 = 30
  • 10 x 4 = 40
  • 10 x 5 = 50
  • 10 x 6 = 60
  • 10 x 7 = 70
  • 10 x 8 = 80
  • 10 x 9 = 90
  • 10 x 10 = 100

Multiplication charts offer numerous benefits for students. They provide a clear, visual way of presenting the multiplication facts, which can help students who learn best through visual aids or repetition. Students can easily refer to the charts while solving problems, allowing them to check their answers and reinforce their understanding. Mastering multiplication is critical for advancing to more complex subjects like division, fractions, and algebra.