TASK: What is a polynomial? Explain what a polynomial is and provide an example. (Look at your flashcards if you need a reminder)
We can combine polynomials by using the four operations (addition, subtraction, multiplication, and division). This lesson will discuss how we add and subtract polynomials. The next lesson will show you how to multiply polynomials. You will learn how to divide polynomials in Unit 4.
The most important thing to remember when working with polynomials is that they are a group of numbers. This means that each polynomial needs to be in parentheses at the beginning of the problem. However, most of the time there are not a lot of like terms to combine within the parentheses. Therefore, after we combine all of the like terms within the parentheses, we need to get rid of the parentheses.
TASK: How do we get rid of parentheses?
Example: Find the sum of
and 
.
The key word "sum" indicates that we have to add these two polynomials, so we can set up our problem like this:
Remember to put the parentheses because polynomials are a group!
Since there are no like terms inside each set of parentheses that we can combine, we have to use the distributive property to get rid of the parentheses. However, there's no number outside the parentheses for us to distribute.
TASK: When there's no number outside the parentheses for us to distribute, what number can we put there? Why does this work?
So we get:
When we distributed the positive 1s to both sets of parentheses, we got rid of the parentheses and everything else stayed the same. Then, we combined the like terms to get out answer.
Be careful when combining like terms because the signs of the numbers can get tricky. The easiest way to remember it is that whatever sign is in front of the term is the sign that goes with it. For example, in the problem above, we combine positive 9 and negative 12 to give us the negative 3 in the answer.
Subtracting polynomials is the same as adding polynomials, except there's one step that's a little trickier. Let's look at the same example from above, but instead of adding them, we're going to subtract them. We set up the problem like this:
We still have no numbers outside the parentheses, so we put the ones:
This is where the problem gets a little trickier because this time instead of distributing a positive 1 to each set of parentheses, we're distributing a positive 1 to the first set of parentheses, but a negative 1 to the second set of parentheses. This is going to change the signs of each term in the second polynomial. So we get:
Take a look at one more example. It follows the same steps, but it looks a little different:
TASK: Explain the steps done in the example problem above.
We can combine polynomials by using the four operations (addition, subtraction, multiplication, and division). This lesson will discuss how we add and subtract polynomials. The next lesson will show you how to multiply polynomials. You will learn how to divide polynomials in Unit 4.
The most important thing to remember when working with polynomials is that they are a group of numbers. This means that each polynomial needs to be in parentheses at the beginning of the problem. However, most of the time there are not a lot of like terms to combine within the parentheses. Therefore, after we combine all of the like terms within the parentheses, we need to get rid of the parentheses.
TASK: How do we get rid of parentheses?
Example: Find the sum of
and .The key word "sum" indicates that we have to add these two polynomials, so we can set up our problem like this:
Remember to put the parentheses because polynomials are a group!
Since there are no like terms inside each set of parentheses that we can combine, we have to use the distributive property to get rid of the parentheses. However, there's no number outside the parentheses for us to distribute.
TASK: When there's no number outside the parentheses for us to distribute, what number can we put there? Why does this work?
So we get:
When we distributed the positive 1s to both sets of parentheses, we got rid of the parentheses and everything else stayed the same. Then, we combined the like terms to get out answer.
Be careful when combining like terms because the signs of the numbers can get tricky. The easiest way to remember it is that whatever sign is in front of the term is the sign that goes with it. For example, in the problem above, we combine positive 9 and negative 12 to give us the negative 3 in the answer.
Subtracting polynomials is the same as adding polynomials, except there's one step that's a little trickier. Let's look at the same example from above, but instead of adding them, we're going to subtract them. We set up the problem like this:
We still have no numbers outside the parentheses, so we put the ones:
This is where the problem gets a little trickier because this time instead of distributing a positive 1 to each set of parentheses, we're distributing a positive 1 to the first set of parentheses, but a negative 1 to the second set of parentheses. This is going to change the signs of each term in the second polynomial. So we get:
Take a look at one more example. It follows the same steps, but it looks a little different:
TASK: Explain the steps done in the example problem above.
ALGEBRA 2 Difference of Two Squares
Before I show you any special guys, you need to be very familiar with some basic perfect squares:
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And some perfect cubes:
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You should know these cold. If my cat comes into you room some night, wakes you out of a dead sleep and yells, "meow meow meow meow!" OK, pretend that he speaks English and yells, "64! What is it? WHAT IS IT?!" Without even thinking, you should yell, "A perfect square! Don't hurt me!"
(Don't worry. My cat isn't allowed out at night.)
OK -- let's go!
Special Guy # 1:
The difference of two squares
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Check it out:
Factor 
Write it as
Check using FOIL -- Believe me?
TRY IT:
| Factor |  |
Here's another one:
Factor 
Rewrite it as squares...
YOUR TURN:
Factor 
What about this guy?
What did we call special guy # 1 ?
The DIFFERENCE of two squares!
is a sum, not a difference... and we do NOT know how to factor him yet! ( Later!)
References:
http://www.coolmath.com/algebra/04-factoring/06-difference-squares-cubes-03
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