Today we're going to talk about rational expressions. A rational expression is just a ratio of two algebraic expressions. So we'll start by looking at a little bit more closely what a rational expression is. And then we'll do some examples looking at the domains of rational expressions.
So let's look at some examples of rational expressions. We know that a rational expression is just the ratio of two algebraic expressions, specifically some polynomial p of x over some polynomial q of x. So it has a polynomial both in the numerator and in the denominator.
So a couple of examples could be 3x squared over x plus 5, or 2x over 3x squared minus 8. So both of these are examples of rational expressions.
And similar to with numerical fractions, with a rational expression the denominator cannot be equal to zero. So when we're looking at a rational expression, the polynomial that's in the denominator cannot be equal to zero.
This entire expression in the denominator cannot be equal to 0, otherwise the entire rational expression is going to be undefined. So if the denominator is zero, the entire rational expression is undefined. So our rational expression is going to have limitations on its denominator, meaning there are values for x that x cannot be, specifically when it will make the denominator equal to zero.
So for this example, x could not be equal to negative 5, because negative 5 plus 5 would be 0. So the domain of this rational expression does not include the value of negative five.
So here's another example of a rational expression. x plus 2 times x minus 3 over x plus 1 times x minus 4. Now I know that there will be restrictions on the domain for whichever values of x will make the denominator equal to 0. So I don't want the denominator equal to 0.
And in this expression the denominator would be equal to 0 if this factor, x plus 1, equals 0, or if this factor, x minus 4, equals 0. So I want to find the values where x plus 1 equals 0 or x minus 4 equals zero. And those will be the values that are not included in the domain.
So I want to find when x plus 1 is equal to 0 and x minus 4 is equal to 0. So here to get x by itself, I'll subtract 1 from both sides. And I find that x is equal to negative 1. And here I'll add 4 on both sides to get x by itself. And I find that x is equal to 4. So the domain of this algebraic expression is going to be all x values not equal to negative 1 and not equal to positive 4.
So here's another example of a rational expression, 7x over x squared minus 3x minus 10. I can find restrictions on the domain by identifying which values of x would make the denominator equal to 0. And to do that, I notice that I can factor this quadratic expression.
My numerator will stay the same. But in the denominator, I know that to factor I can identify two integers that add to give me negative 3. And those same two integers should multiply to give me negative 10. So thinking of integers that multiply to give me negative 10, I notice that negative 5 and positive 2 will multiply to give me negative 10 and also add to give me negative 3. So here I can write this in factored form as x minus 5 and x plus 2.
So now that I have this in factored form, I can more easily identify the restrictions on the domain. I know that if either x minus 5 equals 0 or x plus 2 equals 0, then the whole denominator will be equal to 0.
So I can find what makes x minus 5 equal to 0 and what makes x plus 2 equal to 0. So here I'll add 5 on both sides. And I see that when x is equal to positive 5, the whole denominator will be equal to 0.
Similarly, here I'll subtract 2 on both sides. And I find that when x is equal to negative 2, the whole denominator is 0. So the domain of this rational expression is going to be all x values except when x is equal to 5 or when x is equal to negative 2.
So here's another example of a rational expression, 2x plus 6 over x squared plus 8 x plus 15. I can find the restrictions on the domain for this algebraic expression. But first I notice that both the numerator and the denominator can be factored and written in more simple terms.
So 2x plus 6, 2x and 6 both have a common factor of 2. So I can factor out a 2. And then I'll have as my other factor x and positive 3, because 2 times x will give me 2x. And 2 times 3 will give me 6.
In my denominator I see that this is a quadratic function. So I can factor it by identifying two integers that add to be positive 8 and multiply to be 15. So those two integers would be positive 3 and positive 5. 3 plus 5 gives me 8. And 3 times 5 gives me 15. So here I can write in factored form x plus 3 and x plus 5.
So now that I have both my numerator my denominator factored, I see that they both have a common factor. And so I can cancel this out. Cancel out the common factor in both the numerator and in the denominator. So I'm left with 2 in my numerator and x plus 5 in my denominator.
So to find the restrictions on the domain, I know that my denominator cannot be equal to 0. So I want to find the value that makes the denominator 0. So if x plus 5 is equal to 0, to determine x we'll subtract 5 on both sides. And I see that x is equal to negative 5. So negative 5 is not in the domain of my rational expression.
Even though I canceled out my x plus 3 factor from the numerator and the denominator, there still is a restriction on the domain that comes from this x plus 3 factor being equal to 0. So I also need to set this equal to 0. To determine x, I'll subtract 3 from both sides. And I see that x is equal to negative 3. So I can say that the domain of my original rational expression is going to be all x values, but x cannot be equal to negative 5 or negative 3.
So let's go over our key points from today. A rational expression is a ratio of two algebraic expressions, specifically with a polynomial expression in the numerator and denominator. The expression in the denominator is not allowed to evaluate to zero or else the entire expression is undefined. And because of this, rational expressions have restrictions to their domains or values that the variables may not be.
So I hope that these key points and examples helped you understand a little bit more about rational expressions. Keep using your notes. And keep on practicing. And soon you'll be a pro. Thanks for watching.
a fraction in which the numerator and denominator are polynomials