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(male narrator) In this video,
we will look at adding and subtracting
rational expressions
that already have a common denominator.
We will finally do them
the exact same way we add with numbers,
and that would be to add the numerators...
and keep the denominators.
However, subtracting is the only exception,
where we must first distribute...
the negative into the numerator of the fraction.
Also, be sure to check your answer and not forget to reduce.
Let's take a look at some examples
where we have to do just that.
In this problem,
we notice the denominators are exactly identical.
In this case, we just have to add the numerators--
or combine like terms.
x squared has no like term,
but 4x can be combined with the x to give us plus 5x.
Finally, plus 6 at the end over our common denominator
of x squared, minus 2x, minus 15.
However, before we say this is our final answer,
we will want to make sure we check to see if it can reduce.
To reduce, we must first factor the numerator and denominator.
Using the AC method, we multiply to 6 and add to 5.
This would be 2 and 3,
and because of the 1 in front of x squared,
we can plug those numbers straight into our factors:
x plus 2, times x plus 3.
Over in the denominator, we use the same pattern--
the AC method--multiplying to -15, and adding to -2,
would be 5 and 3, where the 5 is negative.
This give us x minus 5, times x plus 3.
Notice now that we're factored,
we do have a common factor that can divide out.
This leaves us with our final answer:
x plus 2, over x minus 5.
Let's take a look at another example
where we combine the numerators
and then reduce by first factoring.
Again, in this problem,
you notice we have a common denominator.
However, we also have subtraction.
Recall with subtraction, we must first take this negative
and distribute it onto the entire numerator.
This will give us x squared, plus 2x, over 2x squared,
minus 9x, minus 5.
And now, we have plus,
because we've taken care of the negative, -6x minus 5,
over 2x squared, minus 9x, minus 5.
Distributing the negative, you may recall,
simply changes the sign on each term.
Now that we have a common denominator,
we can combine our numerators.
x squared has no like term, but 2x minus 6x is -4x.
And finally, the minus 5 has no like term,
over our denominator: 2x squared, minus 9x, minus 5.
Before we say this is our final answer,
we will want to check to be sure we can reduce.
We must first factor in order to reduce using the AC method.
Multiplying to -5 and adding to -4, we have -5 and +1.
Because of the 1 in front of x squared,
we can factor this to x minus 5, times x plus 1.
In the denominator,
we're multiplying to -10 and adding to -9.
This is done with -10 and 1.
With the 2
in front of x squared,
we're careful to factor
to 2x and x.
The 2x must be multiplied by -5,
and the x by +1,
to give us the -10 and 1.
Notice, we have a common factor
of x minus 5
that can divide out--
leaving our final answer
of x plus 1 over 2x plus 1.
Distributing the negative first, combining terms,
and reducing to find our solution
with common denominators.