Hello wonderful mathematics people. This is Anna Cox from Kellogg Community College Polynomials. We're going to look at the degree of a single term. A polynomial is made-up of the addition and subtraction of multiple terms. If there's only one term, it's called a monomial. If there's 2 terms that are added or subtracted, it's a binomial. If there's three, it's a trinomial, etcetera. So the degree of a term is actually the exponent on the variable portion, or the exponents if there's multiple variables in the single term. So if it's plain old X, it's understood to be to the first power if there is no exponent. If it's a constant, we could think of this constant as being times X to the 0 power because anything to the 0 power is one with the exception of 0 to the zero, which is undefined. So then if we rewrite 3 as 3 * X to the zero, we can see the degree of that term would be 0 because it's the exponent on the variable. If I have 6X, the understood exponent on that X IS17Y to the fifth is going to be a five. Now it gets a little trickier. When we have multiple variables in a single term. All we do though is we add the exponents, so 2 + 9 or 11. So we got that from 2 plus the 9. If we have 3, we just Add all three of them, 12 + 5 + 1, so 18. When we talk about the degree of a polynomial, we look at the degree of each and every individual term and then take whichever single one is the highest. So if all these terms are added or subtracted together, the degree of that polynomial would be 18. So we're going to let you try that one. We have a degree of a polynomial here. The degree of this first term is one. The second term is five. The next term is 11, this next 1:12 and 5:00 and 1:18, and finally zero. So the degree of this whole polynomial would be 18. We don't add up each term. We look at each term individually and then choose the highest. When we combine like terms, we look for the variable portions that look exactly the same. So if I have a single XY here, I can see that this one also is a single XY. So if I have six and three XY's using the distributive property, I can see that I end up with 9 XY's. Then I would look at the next kind of variable. Here's an X ^2 y. I look to see if there's any other X ^2 y S and there is. So there's a -7 -, 11 X squared Y or -18 X squared Y. Then I go to the next variable portion X and AY squared, and I have two of those, so I get 1 -, 12 XY squared or -11 XY squared. That's how we combine like terms. If we want to do descending order, we look at the order of each term, and once we have the order in descending, we want the highest one first. So 5X to the 8th plus 3X to the 4th -7 X squared plus X -, 3. Ascending means going up, descending going down. So once again, we would look at the degree of each and every term, and if we want ascending, we're going up. So -3 + X -, 7 X squared plus 3X to the fourth plus 5X to the 8th. Now in this case I chose the same polynomials. I did that on purpose because I wanted to point out that if it's the same polynomial, the descending and the ascending should be exactly the same, but in opposite order. So if I ended with the -3 here I start with the -3 in the other, the next term is the X, then it would be X, etcetera. Finally, we're going to find values. If we find values, we want to find F of two in this example. So we're going to stick the two in because it's in the parentheses every time I see an X. So F of two means that I'm going to have 3 * 2 to the 4th -7 * 2 ^2 - 3 + 2, so F of two to the 4th 24816. So 3 * 16 - 7 * 4 - 3 + 2. Order of operation says I do the exponents first. Now I'm going to multiply. So 3 * 16 is 48 - 28 - 3 + 2. Then I'm going to add and subtract. So 48 -, 20, it's 2020 -, 3. Seventeen 17 + 2 is 19. Now this really represents on a graph a point. This is saying when X is 2, our F of X or our Y is 19. So this is really a point to 19. Thank you and have a wonderful day.