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(male narrator) In this video,
we will look at multiplying and dividing
rational expressions where we see both operations
in the same problem.
As we do, we remember that dividing really means
we are just multiplying...
by the reciprocal.
When multiplying, we like to reduce first,
but be sure to factor...
before we do any reducing.
Let's take a look at some examples
where we go through that exact process.
Here, we see both multiplication and division
in the same problem.
We notice that the division is going to mean
we need to multiply by that reciprocal.
We now have x squared, plus 3x, minus 10;
over x squared, plus 6x, plus 5;
times 2x squared, minus x, minus 3;
over 2x squared, plus x, minus 6.
And now, we multiply it by the reciprocal,
which will give us: 6x plus 15, over 8x, plus 20.
In order to reduce, we must go through this entire problem
and factor each numerator and denominator.
Let's see what we get as we fill in our factors.
In the first numerator, using the AC method,
we need to multiply to -10 and add to 3.
This is done with 5 and 2, if the 2 is negative.
Because of the 1 in front of x squared,
we know our factors must be x plus 5 and x minus 2.
In the denominator, we multiply also using the AC method--
multiplying to 5 and adding to 6.
We see this is done with 1 and 5.
Because of the 1 in front of the x squared,
we have x plus 1, times x, plus 5.
In the center fraction, we can use the AC method again,
multiplying to -6 and adding to -1.
This is done with 2 and -3.
Because we don't have a 1 in front of x squared,
we will be a little more careful with our factors.
To get 2x squared, we must have 2x times x,
which means the 2x is multiplied by something
to give us an answer of 2.
This is 2x times 1.
In the center, x is multiplied by something
to give us an answer of -3.
This is minus 3.
In the denominator,
we can factor it as well using the AC method:
multiplying to -12 and adding to 1.
This is done with 4 and -3.
Again, we will be careful with our factors,
as we have something in front of x squared.
The only way to get 2x squared is 2x times x,
which means 2x had to be multiplied by something
to get 4: 2x times 2 is 4.
In the center, the x was multiplied by something
to get -3.
It must be -3.
For our last fraction,
all we can do is divide out the greatest common factor.
Here, it's just 3 in the numerator,
leaving us with 2x plus 5.
In the denominator, the greatest common factor is 4,
leaving us with 2x plus 5.
Now that we have gone through
and factored each of these fractions,
we are allowed to do any reducing.
The x plus 5 is in the numerator and denominator,
so is an x plus 1.
We also have a 2x minus 3 and a 2x plus 5,
which show up in the numerator and denominators.
This leaves us with just 3 times x, minus 2,
in the numerator;
over 4, times x, plus 2,
in the denominator.
We can't do any more reducing,
because we have no more matching factors.
By multiplying by the reciprocal and factoring everything,
we can divide out our common factors.
In Part 2 of this video, we will take a look
at solving another similar example.