Hi there! I’m splitting this unit into two parts, over this episode and the next, as this unit is very long and gets complicated quick. Notice how the word polynomial has a prefix of poly. We’ll get into what that means later, but let’s start with what a monomial is.
A monomial can be thought of as just about any number. By definition, a monomial is a number, a variable, or a combination of the two. The following are all examples of monomials:
A binomial is referring to an expression that contains two monomials, such as this:
7x + 5
Keep in mind that a binomial can’t have an like terms, like these:
7x + 9x
This is because we could combine these terms to make 16x, which would be just a binomial.
A trinomial is referring to an expression that contains three monomials, such as this:
7x^2 + 9x + 2
With a trinomial, we will always have at least one monomial with an exponent (in this post, I’ll prelude to an exponent with this symbol: ^). There could be two or three of them, of course, as long as the values of the exponents are different. Here are a few examples:
9x^2 + 6x^8 + 4
34x + 64x^2 -79
80,000x^1/2 + 98 + 23x
Don’t worry about the last one- you won’t need to know how to solve it yet.
Press CONTINUE READING for more of this post!
Right now, I’m just going over some basics that you’ll need to know when we get farther into polynomials. The next step is to find the degrees of a monomial, which are necessary when writing in standard form.
The degree of a monomial is the total value of the exponents. For example, the monomial 3x^2 would have a degree of 2 (since the exponent’s value is 2). Simple, but there are a few other things we have to note. A number on its own, like 3, has a degree of zero. In addition, if we are dealing with a monomial that includes a variable but no visible exponent, we have to remember that the exponent is still one, though we write it without it. For example, the monomial 45x has a degree of 1, since it is technically x^1. You could also have a monomial like 56ab^4. There are two exponents, so we add up their total values (1+4) and find that it has a total degree of 5.
Especially when adding and subtracting polynomials, we want to write these monomials, however many there are, in standard form, which is very different than the one discussed in previous posts. When dealing with polynomials, a polynomial (which is a group of monomials in any amount) is written in standard form when the degrees of the exponents in each monomial are placed in order from greatest to least. Take this polynomial:
7x^2 + 54 + 344^6
And now, here it is in Standard form. You should be able to understand this now:
344^6 + 7x^2 + 54
However… the degree of the polynomial above would be 6, not 8. See, with polynomials, the degree is the greatest value exponent, not a sum of all degrees in the polynomial (while in a single monomial we add together all exponent values, in a polynomial we write it in standard form, thus showing us the greatest degree).
One reason we write polynomials in standard form is to make it very easy when adding and subtracting polynomials. When we add polynomials, it looks like this:
(x^4+7x+2) + (9x^4-9x-6)
From there, we are basically combining the like terms in each polynomial. This can be done either horizontally or vertically. Notice below, I’ve color-coded like terms.
(x^4+7x+2) + (9x^4-9x-6) = 10x^4 – 2x – 4
First off, the above polynomials are already written in standard form, though in some cases you might have to write it into standard form before adding. All I did was I combined the like terms from each side into an expression that couldn’t be simplified any more. Understand?
We use the same method for subtracting, except of course that the + is replaced by a -. This is important though, as you will want to use the distributive property with the negative sign to take the opposite of each monomial in the right polynomial. It’ll look like this:
(x^4+7x+2) – (9x^4-9x-6) = (x^4+7x+2) + (-9x^4-7x-2)
From there, we can do like with addition and combine like terms.
Here’s one thing I forgot to mention: the leading coefficient. This isn’t directly involved in adding and subtracting, but the leading coefficient is the coefficient directly next to the variable with the highest degree in a polynomial. When a polynomial is written in standard form, this is very easy to identify.
6x^2 + 8. The leading coefficient would be 6.
I am aware that I am not very good at explaining things, so let’s review before we move farther:
- A monomial is any number, variable, or a combination of the 2 (through multiplication only- cannot be a sum)
- A polynomial is a combination of any amount of monomials
- The degree of a monomial is the sum of all exponent values in that specific monomial
- Binomials, Trinomials, etc. are all examples of polynomials
- The degree of a polynomial is the greatest degree of a specific monomial in a polynomial
- An expression is in standard form when the degrees of each monomial are written in order from greatest to least
- Polynomials are added by combining like terms from each side to result in a single polynomial. Note: If there is a term that does not, have any like terms, that term is left alone and placed appropriately in standard form, which the final expression should always be written in.
- Polynomials are subtracted by combing like terms form each polynomial after distributing the negative sign to the right-side polynomial. If there is a monomial that does not have any like terms on the left side, it is left alone and placed appropriately in Standard form. On the right side: it’s opposite is placed appropriately in Standard form.
- The leading coefficient is the coefficient of the monomial with the greatest-value degree in a polynomial.
In the next edition, I’ll cover multiplication of polynomials (which is a task in itself) and an intro to factoring polynomials. Luckily for you (but unluckily for me) we’re really behind in Math class, so we still have 3 units to get through before the summer. While the next edition will cover some more on polynomials, it will be the last edition before my June break. After that, I’ll have plenty of new material (including finishing up polynomials) to share with you over the summer. I’m glad that I was able to squeeze so much material into this one post- this is how I want my It All Adds Up posts to be.
Below, I’ve provided a link to the Quizlet study set for this post that you can access by clicking on the Quizlet icon.
Thank you so much for reading!
Click the Quizlet icon to access the study set for today’s lesson.