what do you call the product of rational expression?
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what do you call the product of rational expression?
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Answer:
We now need to look at rational expressions. A rational expression is nothing more than a fraction in which the numerator and/or the denominator are polynomials. Here are some examples of rational expressions.
Step-by-step explanation:
The last one may look a little strange since it is more commonly written
4
x
2
+
6
x
−
10
4x2+6x−10. However, it’s important to note that polynomials can be thought of as rational expressions if we need to, although they rarely are.
There is an unspoken rule when dealing with rational expressions that we now need to address. When dealing with numbers we know that division by zero is not allowed. Well the same is true for rational expressions. So, when dealing with rational expressions we will always assume that whatever
x
x is it won’t give division by zero. We rarely write these restrictions down, but we will always need to keep them in mind.
For the first one listed we need to avoid
x
=
1
x=1. The second rational expression is never zero in the denominator and so we don’t need to worry about any restrictions. Note as well that the numerator of the second rational expression will be zero. That is okay, we just need to avoid division by zero. For the third rational expression we will need to avoid
m
=
3
m=3 and
m
=
−
2
m=−2. The final rational expression listed above will never be zero in the denominator so again we don’t need to have any restrictions.
The first topic that we need to discuss here is reducing a rational expression to lowest terms. A rational expression has been reduced to lowest terms if all common factors from the numerator and denominator have been canceled. We already know how to do this with number fractions so let’s take a quick look at an example.
not reduced to lowest terms
⇒
12
8
=
(
4
)
(
3
)
(
4
)
(
2
)
=
3
2
⇐
reduced to lowest terms
not reduced to lowest terms ⇒ 128=(4)(3)(4)(2)=32 ⇐ reduced to lowest terms
With rational expression it works exactly the same way.
not reduced to lowest terms
⇒
(
x
+
3
)
(
x
−
1
)
x
(
x
+
3
)
=
x
−
1
x
⇐
reduced to lowest terms
not reduced to lowest terms ⇒ (x+3)(x−1)x(x+3)=x−1x ⇐ reduced to lowest terms
We do have to be careful with canceling however. There are some common mistakes that students often make with these problems. Recall that in order to cancel a factor it must multiply the whole numerator and the whole denominator. So, the x+3 above could cancel since it multiplied the whole numerator and the whole denominator. However, the
x
x’s in the reduced form can’t cancel since the
x
x in the numerator is not times the whole numerator.
To see why the
x
x’s don’t cancel in the reduced form above put a number in and see what happens. Let’s plug in
x
=
4
x=4.
4
−
1
4
=
3
4
4
−
1
4
=
−
1
4−14=344−14=−1
Clearly the two aren’t the same number!
So, be careful with canceling. As a general rule of thumb remember that you can’t cancel something if it’s got a “+” or a “-” on one side of it. There is one exception to this rule of thumb with “-” that we’ll deal with in an example later on down the road.