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(male narrator) In this video,
we will look at reducing rational expressions
made up of several polynomials.
We remember that to reduce, we divide out...
common factors.
What this means to us is we must first factor the polynomial
before we are allowed to do any reducing--
just as with factoring fractions.
In this problem,
we cannot simply jump to reducing the terms.
We never will reduce terms.
Instead, we must first factor the numerator and denominator.
In the numerator, using the AC method,
we need to multiply to -6 and add to 5.
This is possible with 5 and 1...
I'm sorry, with 6 and 1, if the 1 is negative.
We can start building our factors then.
The only way to get 2x squared is 2x times x.
We had to factor a 2 out of one of our numbers.
Clearly, the 2 can factor out of the 6, giving us a +3.
Factoring the x out of -1x leaves us just -1,
and the numerator is factored.
Similarly, we must factor the denominator.
Using the AC method, we multiply to 4 and add to -5.
We see this is possible with -4 and -1.
We will again create our factors,
keeping in mind FOIL.
The only way to multiply to 2x squared is 2x times x.
On the outside, 2x had to be multiplied by something
to get an answer of -4.
This would be 2x times -2.
In the center, the x must have been multiplied by -1.
Now that we have factored our expression,
we are allowed to reduce.
Notice, we have a common factor
in both the numerator and denominator of 2x minus 1.
This factor can divide out,
leaving us with just x plus 3, over x, minus 2,
as our reduced rational expression.
Let's take a look at another example
where we have to factor first before we're allowed to reduce.
In this problem,
we can't jump right to reducing the 9x squareds
as a result of the pluses and minuses in the problem.
We must first factor that numerator.
Using the AC method, 9 times 25 is 225.
We also want to add to -30.
After some practice, we find this will be -15 and -15.
Notice, those two numbers match.
When they match,
we know it's one of our special factoring shortcuts.
It's going to be something squared.
Taking the square root of the 9x squared to get 3x,
the square root of the 25 to get 5,
and the sign from the middle
will complete factoring the numerator.
In the denominator,
we will need to factor the 9x squared minus 25.
Notice, it's a binomial.
We have a difference of squares.
Using our difference of squares shortcuts,
we know it's going to factor
to the sum and the difference of the square roots.
The square root of 9x squared is 3x,
and the square root of 25 is 5: 3x plus 5, times 3x, minus 5.
Now that it's factored,
we are finally allowed to reduce.
We have a common factor in the denominator and the numerator
of 3x minus 5.
As we reduce it out of the denominator,
we also notice there were two to begin with
and now one left over in the numerator.
We leave one behind in the numerator, 3x minus 5;
over our denominator, 3x plus 5.
And now, the expression is simplified.
We cannot reduce the 3s and 5s
as a result of the addition and subtraction.
This is completely simplified by factoring.