LESSON 2:THE MULTIPLICATION OF POLYNOMIALS

A-SSE.A.2

A-APR.C.4

Opening Exercise

5 minutes, with explanation (2 slides)

Show that using an area model. What do the numbers you placedinside the four rectangular regions you drew represent?

(Reminder: An area model is when you do multiplication by creating a rectangle whose sides are the factors and whosearea is the product of those factors)

EXPLANATION OF THE OPENING EXERCISE

Show that using an area model. What do the numbers you placed inside the fourrectangular regions you drew represent?

An area model suggests that we should draw a rectanglewhose sides are the factors, but it wants us to use instead of and respectively.

Then we want to find the area of each section.

The total of all the smaller areas must be the total area of the shape:

20

0

8

7

400 160

140 56

EXAMPLE 1

If we can use the above method to multiply two quantities represented by the addition of numbers, then maybe it can also extend to when we don’t necessarily know what the numbers are.

How is the expression similar to ?

If we replace with , we get back to the question from the opening exercise!

At this point, it will help to think of polynomial multiplication taking place inside of an area model (even though technically it cannot always be expressed like this; for example, what if Both and would be negative, but area cannot be negative).

Still, if we visualize polynomial multiplication like this, our calculations will be quick, easy to follow, and structured in a way to combine all relevant information.

EXAMPLE 1

So let’s do that multiplication!

𝑥+8𝑥

+¿7

𝑥2 8 𝑥

7 𝑥 56

𝑥2

56

(𝑥+8 ) (𝑥+7 )=𝑥2+15 𝑥+56

DRAWING RELATIONSHIPS

How is the in Example 1 related to the from the Opening Exercise?

Hint: What would have to be for us to get an answer of ?

One step further: If we wanted to know the answer to , what could we do instead of calculating it directly?

Answers to these on the last slide.

MULTIPLICATION WITHOUT A TABLE

Remember, when we want to multiply two factors that are composed of many parts, we just need to make sure we distribute each part to each other part. The area method (table) does a good job of visualizing this, but you don’t need an area model to get the same result:

Let’s start with the same problem

𝑥+8(𝑥+7)We’re now going to treat as a single entity. *

* This is very common to do in higher levels of

Algebra. As long as things are able to be

grouped, you can choose to group

them, and then work with the group as if it

were a single variable.

() Distribute over the remaining terms𝑥+8

(𝑥+8 ) 𝑥+(𝑥+8 )7 Here we just gave the to the and the

𝑥2+8 𝑥+7 𝑥+56 Distribute back (give the to the , and the to the )

𝑥2+15 𝑥+56 Again, we get the same answer.

IMPORTANT QUESTIONS

What property did we repeatedly use to multiply the binomials?

How is the tabular method similar to the distributive property?

Does the table work even when the binomials do not represent lengths? Why/why not?

(Answer these in your notebook)

EXERCISE 1 AND 2

1. Use the tabular method to multiply and combine like terms.

2. Use the tabular method to multiply and combine like terms.

DO THESE!

EXERCISE 1 AND 2 (ANSWERS)

Not this time! Check them in class with me / your classmates!

EXAMPLE 2

Find the products of the following few expressions. Can you generalize what is happening? (You can use either the distributive property or the tabular method, whichever you find most comfortable).

GENERALIZING THE PATTERN

After doing Example 2 a few times, you may begin to see that the product of two polynomials, one in the form of and the other being a sum of monomials whose exponents are counting down (yeah, a mouthful, but that’s how to describeit) is seemingly going to be equal to .

There’s actually a mathematical way to describe this, and it’s what we do when we say “generalize this statement.”

To generalize something means to put it into terms for which it could apply to ANY statements of that form.

In this case, the pattern suggests the following happened:

Let’s break that down a bit on the next slide.

GENERALIZING EXAMPLE 2

(𝑥−1 ) (𝑥𝑛+𝑥𝑛−1+…+𝑥2+𝑥+1 )=𝑥𝑛+1−1

This part seems to be in all of these.“A difference of from .”

In this case, we can take this to mean that can be raised to any naturalnumber (integer ) power, and this identity will hold. Simply pick a numberfor , and plug it in!

For the rest of this, we seem to be adding every term,and the exponent of those terms keep going downby 1 every time we do an addition.

Remember, these terms are still with exponents too! ( and )

All of the middle termscancel each other outand we’re left with justthe first and the lastterms being multiplied:

(𝑥 ) (𝑥𝑛)=𝑥𝑛+1

DO YOU REMEMBER THIS?

You’ve done this before! Well, kind of. You stopped after .

That’s right; we’ve just generalize the Difference of Two Squares (DoTS) “formula”.

This is why I hate calling a lot of things formulas…there’s usually so much more to them, and they can be generalized tolarger pieces. We now have a way of figuring out how to move between items of the form and

Now let’s go one step further…

INTRODUCTION TO EXERCISE 3-4

Take a moment and attempt to multiply the following two sets of expressions:

INTRODUCTION EXPLANATION

Above we have the expansion of the factors for the difference of two squares and the difference of two cubes, respectively.

EXERCISE 3

Multiply using the distributive property and combine like terms. How is this calculation similar to example 2?

EXERCISE 3 EXPLANATION

Multiply using the distributive property and combine like terms. How is this calculation similar to example 2?

Start by distributing one expression into the other (or using the tabular method) to get:

Which simplifies into… 𝑥4−𝑦 4

At this point, hopefully it’s becoming obvious that “FOIL” or “double bubble” is no longer an effective wayto multiply polynomial expressions. These multiplications are ALL just repeated use of the distributive property.

EWE: Each With Each – just make sure that each term in one polynomial is multiplied with each in the other

EXERCISE 4

Take some time with a partner (or if you’re at home, get this set up and save it for small group discussion) to do the following:

Expand the following expressions:

ANSWERS TO EXERCISE 4

Very similarly to the other few examples, all of the middle terms appear to cancel out, and we wind up with the product ofthe first and last terms when we have a difference of two terms multiplied by the sum of those two terms.

(𝑥2− 𝑦2 ) (𝑥2+𝑦 2 )=𝑥4− 𝑦4

LET’S GENERALIZE!

We’re going to use a table to figure out what happens when we have the general form.

Remember, when we generalize, we want to keep the same structure, but replace the things that are changing with a variable separate from the ones we may be dealing with.

𝑥𝑛 − 𝑦𝑛

Remember, it’s okay to group the sign with the

next monomial.

𝑥𝑛

+¿𝑦 𝑛

𝑥2𝑛 −𝑥𝑛 𝑦𝑛

𝑥𝑛 𝑦𝑛 − 𝑦2𝑛𝑥2𝑛

0 𝑥𝑛 𝑦𝑛− 𝑦2𝑛

for all

PROBLEM SET (PAGE 1 OF 2)

Problem Set assignments belong in your “WORKING” book

Please label this “Assignment ”

PROBLEM SET (PAGE 2 OF 2)