Now that we've seen the basics of polynomials, a lot of problems will ask you to simplify polynomials and write them in a specific form called standard form. That's what I'm going to show you how to do in this video. I'll show you how to write polynomials in standard form. And, really, it just all comes down to the order of the terms in the polynomial. Let's go ahead and take a look here. So, what standard form actually means is that all the terms of your polynomials should be written in descending or decreasing order of exponents. What does that mean? If you look at this expression or this polynomial, 3x2+5x+4, notice how the exponents actually keep decreasing. First, we have an x2. Then we have a 5x1. Right? There's an invisible one there. And then you have +4. And one way you can think of 4, remember, is that you have x0, and x0 is just 1. So it's kind of weird, but, basically, you can see here that the order of the exponents keeps on decreasing: 2 to 1 to then 0. And the other thing is that all like terms have to be combined. In other words, you can't simplify the expression any further. So, if you look at this expression, I can't combine anything with the 3x2 or the 5x or the 4. So it turns out that this expression is already in standard form. Alright? A couple of other things that you should know. One thing is called the degree of the expression, and that's basically just the highest exponent of the variable that you see in the polynomial. So, in other words, the highest exponent that we see that's attached to a variable is the x2. It's the 2. So that means that this is called a second order or a second-degree polynomial. And last but not least, there's just a new vocabulary word. We know that the numbers not attached to variables or by themselves are called constants. We know numbers in front of variables multiplying them are called coefficients, and there's a special name for the number that goes way out in the front that gets attached to the variable of the highest exponent, and that's called the leading coefficient. It's like the number that leads the entire expression. Alright? That's a leading coefficient. That's basically it. So lots of problems are going to ask you to now write expressions in standard form, and that's what we're going to do in this problem. We're going to identify the degree and the leading coefficients. Let's get started here with this first expression. So in other words, we have 12x1 over here plus x3. So we have to write this in standard form, and that means that we have to write it in decreasing order of exponents or descending order. So we have an x3 and an x1. That's backwards, so I just have to flip the two expressions. And when you move expressions around, just be very careful what happens to the signs. In other words, I'm going to rewrite this as x3+12x. Alright? Because it's basically just the x1. So does this have decreasing order of exponents? Yes. It does. And are all the like terms combined? I can't combine anything with the x3 and the one-half x, so this definitely is a simplified expression. So it's in standard form. So what's the degree? The degree is really just the highest exponent that gets attached to a variable, and that one is 3. So in other words, the degree over here is 3. And what about the leading coefficients? Well, the leading coefficient is the number that gets attached to that variable with the highest exponent. And over here, what you'll see is that, basically, there wasn't a variable, but remember, there's always kind of like an invisible one there if you don't see a number. So the leading coefficient in this case is actually just 1. Okay? So let's take a look at this expression; it was a little bit more complicated. There are more terms. Notice that there are some terms with x. There are some with x2, and there are actually just some constants in here. So I have to write them in descending order, but be very careful when you do this because you basically have to keep track of the signs. So, in other words, I have a -3x2 and an x2. So I'm going to move that. I have a 5x and a 2x. I'm going to move that, and I have a -7. So I want to make sure that all the x2s go first. So this is going to be -3x2+x2. Then I'll have the +5x+2x, and then I have the -7. So when you pick these numbers up around and move them, remember that you're always doing this with the sign that goes in front of them. Alright? So now what happens is we have descending order of exponents. I have exponents 2 and then 1, and then I have 0 over here. So this definitely has descending order, but it's not as simplified as it could be because I could still combine all the like terms. So that's what I have to do in the second step. So if I combine this expression over here, -3x and 1x, it's kind of like -3apples and 1apple. This actually just becomes -2apples. What happens to the 5 and the 2? That becomes the 7x, and then the -7 just becomes -7. So now all the like terms have been combined, and this definitely now is in standard form. So what's the degree of this polynomial? What's the highest variable highest exponent of a variable that we see? It's just the 2 over here. Alright? So in other words, we have 2. And then what's the leading coefficient? What's the number that goes in front of that term? It's actually just this -2 over here. So that's the -2. Alright? So, really, that's all. That's it for this one, folks. Let me know if you have any questions, and we'll see you in the next video.
Table of contents
- 0. Review of Algebra4h 16m
- 1. Equations & Inequalities3h 18m
- 2. Graphs of Equations43m
- 3. Functions2h 17m
- 4. Polynomial Functions1h 44m
- 5. Rational Functions1h 23m
- 6. Exponential & Logarithmic Functions2h 28m
- 7. Systems of Equations & Matrices4h 6m
- 8. Conic Sections2h 23m
- 9. Sequences, Series, & Induction1h 19m
- 10. Combinatorics & Probability1h 45m
0. Review of Algebra
Polynomials Intro
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