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Lesson 116: Working with Polynomials (Part 1 of 2)
In the last lesson we learned about monomials, now we'll learn about polynomials, which are just a sum or difference of monomials.
First, some definitions. A monomial is a special type of polynomial that just has one term, such as 17, or 3x. A binomial is a polynomial with two unlike terms, such as 3x - 4, or 7x2 + 3x. A trinomial is a polynomial with three unlike terms, such as 9x2 + 3x - 5. The prefix poly just means many. A polynomial is just a sum and/or difference of one or more monomials. For example, 7x3 + 2x4 - 5x - 7 is a polynomial, but it doesn't have another special name like the ones above do.
When we work with a monomial with only one variable, like the examples above, the degree of the polynomial is the value of the highest exponent in the polynomial. For example, the degree of 9x2 + 3x - 5 is 2. The degree of 4x - 3 is 1, because the x is really x1. The degree of a monomial which is just a constant is 0. There is no variable raised to any power.
Simplifying Polynomials
A common task is to simplify polynomials. All we do is look for like terms that we can combine. For example, if we have 9x2 + 3x - 5 - 7x, we can combine the x terms. We have to be very careful with our signs. What we have is 3x - 7x, giving us -4x. Our simplified polynomial would be 9x2 - 4x - 5. We can only combine like terms in this fashion. For example, we don't combine 9x2 with -7x. One is a squared term, and the other isn't.
Adding Polynomials
Another task is to add and subtract polynomials. This is
similar to what we did above, but we have to be very carful with our
signs. Let's try this: (7x2 + 4x - 9) + (8x2
- 5x - 8). What we do is combine both polynomials into one big
one. We have:
7x2 + 4x - 9 + 8x2 - 5x - 8. Look at the
signs. Since we're adding the second polynomial, we just leave
the signs as they are. Later we'll see how this is different
when you subtract.
Now we have to combine whatever like terms we can. The squared terms can be combined to get 15x2. For the x terms, we'll do 4x - 5x, to give us -1x, which we can write as -x. For the constants, we have -9 - 8. Be very careful. Our answer is -17. Combining all this together, we have15x2 - x - 17.
Subtracting Polynomials
Let's try subtracting polynomials. Students have a lot of trouble with this, so pay close attention. Let's do the same example as above, but we'll subtract instead of add. We have (7x2 + 4x - 9) - (8x2 - 5x - 8). There are few ways to think about what we're going to do. The way I like to think about it is that we need to distribute the minus sign over every term in the second polynomial. That means we're multiplying each term in the second polynomial by -1, which means that the sign of each term will be reversed.
Another way to think of it is that we're going to subtract each term in the second polynomial, but when we subtract a negative, we actually add. Remember that earlier lesson. The problem becomes 7x2 + 4x - 9 - 8x2 + 5x + 8. Make sure that you see what was done with the signs. Now we combine like terms, being very careful with signs. We end up with -x2 + 9x -1. Make sure that you understand how we got that answer.
You'll have much more practice with this in later lessons, and you'll learn more in the next lesson. Make sure that you fully understand everything that was presented in this lesson.
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