Multiplying Polynomials 2

Contributor: Mason Smith. Lesson ID: 11550

Just when you get the hang of multiplying binomials bigger ones start multiplying! That's OK; there's a method that will allow you to dive in and FOIL those big problems. Watch these videos and learn!

categories

High School, Middle School

subject
Math
learning style
Visual
personality style
Otter
Grade Level
Middle School (6-8), High School (9-12)
Lesson Type
Quick Query

Lesson Plan - Get It!

Audio:

OK, if you don't get it, read on to find out how it relates to multiplying polynomials!

If you have completed the first three lessons in the Polynomials series, you have covered multiplying monomials, and multiplying a monomial by a binomial. If not, please go to Related Lessons in the right-hand sidebar.

Beware: Something funny happens when you start multiplying larger polynomials together! In this lesson, you will learn how to multiply two binomials, as well as how to multiply polynomials together.

To multiply larger polynomials, the distributive property is used to solve the equation by distributing the first term, then multiplying by all terms in the parenthesis.

Note that the second term includes the sign (+ or -) between the two polynomials.

For example: (x + 3)(x + 2)

    1. Distribute x + 3:
      • x(x + 2) + 3(x + 2)
    2. Distribute again inside the parenthesis:
      • (x * x) + (x * 2) + (3 * x) + (3 * 2)
    3. Multiply together:
      • x2 + 2x + 3x + 6
    4. Simplify to get your final answer:
      • x2 + 5x + 6

This seems pretty complicated and difficult, right?

Well, when multiplying two binomials together, you can use a trick called the FOIL method to multiply each term in the first parenthesis by each term in the second. When you multiply two binomials, you multiply:

      1. the First terms of each parenthesis together,
      2. the Outside terms,
      3. the Inside terms,
      4. the Last terms.

Please note: The FOIL method only works when you are multiplying two polynomials.

For examples on how to use the FOIL method, take a few minutes to watch Use FOIL Method to Multiply Polynomials Algebra from TabletClass Math:

 

For extra help, watch Algebra - FOIL Multiplying Binomials by yaymath:

 


Okay, after all that FOILing, you must be super-excited to try it for yourself, but there's one last trick to learn before you dive into the practice of the FOIL method.

Any binomial written as (x + 2)2 can be rewritten as the same binomial times itself using the exponent to indicate the number of times. If your exponent is 2, then you would write the example as (x + 2)(x + 2). If you had (x + 2)3, you would write that as (x + 2)(x + 2)(x + 2). After that, you would just solve as you would any other problem.

Give these problems a try using the FOIL method, and share your answer with a parent or teacher. Drag the correct answer into the answer space:

Whenever you multiply larger polynomials together, you follow the same idea as multiplying binomials.

Begin by multiplying each term by the opposite side, but you'll soon notice that the FOIL method does not work, just like in the comic at the start of the lesson. You have to be extremely careful that you don't miss a term, or else you will end up with a wrong answer. So, what do you do with that extra term?

Take a look at these examples:

(5x + 3)(2x2 + 10x - 6)

  1. Distribute the 5x + 3:
    • 5x(2x2 + 10x - 6) + 3(2x2 + 10x - 6)
  2. Distribute again:
    • (5x * 2x2) + (5x * 10x) + (5x * -6) + (3 * 2x2) + (3 * 10x) + (3 * -6)
  3. Multiply terms together:
    • 10x3 + 50x2 - 30x + 6x2 +30x -18
  4. Rearrange like bases together:
    • 10x3 + 50x2 + 6x2 -30x +30x - 18
  5. Combine like terms for your final answer:
    • 10x3 + 56x2 -18

(x - 3)3

  1. Rewrite as binomials:
    • (x - 3)(x - 3)(x - 3).
  2. Multiply the first two using FOIL:
    • [(x * x)(-3 * x)(-3 * x)(-3 * -3)](x - 3).
  3. Simplify:
    • [x2- 3x - 3x + 9](x - 3).
  4. Combine like bases:
    • [x2- 6x + 9](x - 3).
  5. Use thecommunicativeproperty of multiplication to rewrite:
    • (x - 3)[ x2- 6x + 9].
  6. Split terms:
    • x(x2- 6x + 9) -3(x2- 6x + 9).
  7. Distribute:
    • (x * x2) + (x * - 6x) + (x * 9) + (-3 * x2) + (-3 * - 6x) + (-3 * 9).
  8. Multiply together:
    • x3- 6 x2+ 9x - 3 x2+ 18x – 27.
  9. Rearrange so like terms are together:
    • x3- 6 x2- 3 x2+ 9x + 18x – 27
  10. Combine like terms for your final answer:
    • x3+ -9x2+ 27x –27

Give these problems a try and share your answer with a parent or teacher. Drag the correct answer into the answer space:

Once you feel comfortable multiplying using FOIL and multiplying larger polynomials, move onto the Got It? section to practice your skills.

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