Multiplying Polynomials and Binomials

Lesson Plan - Get It!

Audio:

To multiply larger polynomials, the distributive property is used to solve the equation by distributing the first term and multiplying it 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.

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 in this order.

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 multiplying two polynomials.

For examples of how to use the FOIL method, take a few minutes to watch the video below.

For extra help, watch this next video as well.

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 solve as you would any other problem.

Try these problems using the FOIL method. 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 not to miss a term, or you will have 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 the communicative property 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

Try these problems. Drag the correct answer into the answer space.

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