Hi, and welcome. My name is Anthony Varela. And today I'm going to introduce rational expressions.
So we'll talk about rational expressions, define them, and take a look at some examples. We'll also talk about rational functions. And specifically we'll be looking at the domain because there are some restrictions that we have to consider when talking about the domain of rational functions.
So first, what is a rational expression? We'll, it's a fraction in which the numerator and the denominator are polynomials. So let's take a look at some examples of rational expressions.
So we see a fraction. And in both the numerator and the denominator, we have a polynomial expression. So I'd like to point out that you might be seeing one term in either the numerator or the denominator. But it's all considered a polynomial expression. So these are all valid rational expressions.
So a rational expression can also be called an algebraic fraction because we see a fraction with variables. So That's an algebraic expression.
Well, as a function, we can generally express a rational function as f of x equaling p of x over q of x. So it's a fraction again where the numerator is a polynomial function and the denominator is a polynomial function.
Now because this is a fraction, we know that the denominator cannot equal 0. So with numeric fractions, we know it doesn't make sense to say 4/0. That's not allowed so with our algebraic fractions or our rational expressions, our denominator is not allowed to equal 0.
So for example in the basic rational expression 1/x, x is not allowed to be 0. So that brings us into a discussion on the domain of rational expressions and the domain of rational functions. It's all x values such that the denominator does not equal 0.
So a domain restriction then would be your x values that makes the denominator equal to 0. So let's look at this rational expression. We have x plus 2 times x minus 3, all over x minus 2 times x plus 4. What are the domain restrictions to this rational expression?
Well, we're going to focus on the denominator. So we don't really care what the numerator is. We just need to find x values that make the denominator equal 0.
So notice that this is a quadratic in factored form. So we have one factor, x minus 2, and another factor, x plus 4. So we can set each of those factors equal to 0 and solve for x pretty easily.
So our x values then are x equals 2 and x equals negative 4. These are the x values that make our denominator equal to 0. So we would say then that the domain is all x values such that x does not equal negative 4 or 2. It can equal anything else-- just not these two values.
Well sometimes it's not as easy when they're not written out in factors. So if we have this rational expression, x cubed plus 4x squared plus 5x plus 2, all over x squared minus x minus 2, once again, I don't really care what the numerator is. I'm focusing just on my denominator, because that's what will help us find our domain restrictions. I need to set my denominator equal to 0 and solve for x.
Well, this expression can be written in factored form. And notice, it's a quadratic expression. I can factor this.
So I'm looking for two integers that when I multiply them together I get negative 2. Adding them together, I get a negative 1. So that would be 1 and negative 2.
So in factored form, this is x plus 1 times x minus 2. We can FOIL this to confirm that.
So now they have it in factored form, I could set each of my factors equal to 0 and solve for x. So when x equals negative 1, my denominator equals 0. When x equals 2, my denominator equals 0. So the domain for this expression then is all x values such that x does not equal negative 1 and x does not equal 2.
Let's take a look at another rational expression. This one's interesting because what I'm tempted to do or what I really should do is simplify this as much as possible. But there's something I need to consider once I find my domain restrictions.
So I notice that in both the numerator and denominator, I see a factor, x minus 1. So because that is a factor on the top and the bottom, that cancels out. So I can simplify this to just x plus 1 times x plus 2 over x plus 3.
So now I'm taking a look at my denominator, x plus 3. I know that all x values such that x does not equal negative 3. But that's not all-- I do have to return to my original expression. I know that x can also not equal positive 1 because that would make x minus 1 equals 0. And that appears in my original denominator. So I have to add that to my restriction.
So always check the original expression when you cancel out factors. Though, they'll still represent restrictions to what x can and cannot be.
So let's review our introduction to rational expressions. Well, a rational expression is an algebraic fraction. So we have a polynomial numerator and a polynomial denominator. As a function, we can say that f of x equals p of x over q of x. So a polynomial function in the numerator, a polynomial function in the denominator.
For a domain, these are values of x that x is allowed to take on. So we're looking for all x values that make the denominator non-zero. So we would just then solve for the denominator equaling 0.
And always check the original expression. This is important when you cancel out common factors in your numerator and denominator. So still need to be considered when eliminating possible values for x.
So thanks for watching this tutorial on an introduction to rational expressions. Hope to see you next time.