1. MA 1128: Lecture 13 – 10/27/14
Rational Expressions

2. Rational Expressions
A rational expression is a fraction where the top and bottom are (or can be written as) polynomials.
For example,
is a rational expression.
We’ll often want the polynomials to be factored.
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3. Recognizing Factors
When dealing with rational expressions, it is important to be able to recognize the factors of the two polynomials.
For example, when you look at
(x + 2)(x – 1)(x2 + 2x – 3),
the first thing you should see is that there are three things being multiplied together.
We’ll say that there are three factors.
In (x2 + 7)(x – 3), there are two factors.
In (x2 + x + 1), there is only one factor.
Looking at the individual factors, the most basic things are the terms.
In x2 + x + 1, we have three things being added.
That is, there are three terms.
A factored polynomial is made up of factors, which in turn, consist of terms.
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4. Factors (cont.)
In 2(x + 1)(x – 1), there are three factors, 2, x + 1, and x – 1.
This is equal to 2(x2 – 1), which is now written with two factors.
And it is also equal to 2x2 – 2, which has only one factor.
I think that it is useful to think of exponents as repeated multiplication in this setting,
so x2 has two factors, x and x.
Along these lines, (x – 1)3 has three factors, x – 1, x – 1, and x – 1.
You may occasionally see something like 2x2 – (x+1)(x2 + 2).
This has only one factor. That subtraction is the first thing you should see.
(Note that the first term has 3 factors, and the second term has 2 factors)
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5. Examples 6x has 2 factors, 6 and x. 23x has 3 factors, 2, 3, and x.
9x4 has 5 factors, 9, x, x, x, and x.
32x4 has 6 factors, 3, 3, x, x, x, and x.
6x2(x+1)(x2 – 4) has 5 factors, 6, x, x, (x + 1), and (x2 – 4).
x2 + (x + 1)(x – 1) has only 1 factor.
When you look at an expression, you should practice seeing how things break apart. (This happens in reverse of the order of operations.)
In this last example, that addition is the weakest link, so it breaks there first into two terms.
The two terms each break at the multiplication into two factors.
Two of the four factors break into two terms.
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6. Practice Problems Be sure you understand the last few slides!
As each expression is written, how many factors are there?
(x + 1)(x – 2).
x2 + 4x + 4.
(x2 + 4x – 2)(x – 1) (x + 13).
3x2(x – 2).
7x2 + 4x.
x(7x + 4).
Answers:
1) 2; 2) 1; 3) 3; 4) 4; 5) 1; 6) 2.
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7. Canceling Factors
I’m going on about factors, because these are the source of a lot of errors in simplifying rational expressions.
Let’s look at regular fractions first.
Remember that multiplying fractions is easy.
You just multiply the tops and multiply the bottoms.
This gives us a way to simplify fractions.
Of course, we usually just write
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8. We Can Only Cancel Factors!
Something seems to make us want to do very wrong things like
BAD!!
BAD!!
This is clearly wrong, since
and
Remember: You can only cancel factors!!!
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9. Examples
GOOD
GOOD
BAD!!
BAD!!
GOOD
GOOD
GOOD
GOOD
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10. Practice Problems
Simplify the following rational expression by canceling factors.
Answers:
1) [x+2]/[x-1]; 2) [x]/[x-3]; 3) [x-4]/[x+3].
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11. More Simplifying We know enough to factor things like 2x3 + 2x2 – 12x.
Factor out a 2x: = 2x(x2 + x – 6)
Then factor the trinomial: = 2x(x – 2)(x + 3)
In simplifying a rational expression, the first thing to is to factor.
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12. Practice Problems
In the following rational expressions, factor and cancel common factors.
Answers:
1) [x-2]/[2(x-3)]; 2) [x-2]/[2x^2(x+5)]; 3) [x^2(x-1)]/[x+1].
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