Tip:
Highlight text to annotate it
X
(male narrator) Now we have a second problem
which requires both multiplying and dividing
in the same problem.
Again, we will solve it by first multiplying
by the reciprocal of the fraction that's divided.
When we do, we have x squared,
minus 1, over x squared, minus x, minus 6;
times 2x squared, minus x, minus 15;
over 3x squared, minus x, minus 4.
And now, we multiply by the reciprocal:
3x squared, plus 2x, minus 8;
over 2x squared, plus 3x, minus 5.
Before we're allowed to do any reducing,
we're going to need to factor each numerator and denominator,
so that we can divide out common factors.
In the numerator of the first fraction,
you notice we have a difference of squares.
This will always factor
to the sum and difference of the square roots:
x plus 1, times x, minus 1.
In the denominator, we can use the AC method,
multiplying to -6, adding to -1.
This is done with 2 and 3, if the 3 is negative,
and because we have a 1 in front of x squared,
we can use those numbers in our factors:
x plus 2, times x, minus 3.
In the center, we're multiplying to 2, times -15, or -30;
and adding to -1.
This will be done with 6 and 5, if the 6 is negative.
Because we have a 2x squared,
we need to be a little more careful as we build our factors:
2x squared is 2x, times x.
On the outside, 2x is multiplied by something to get -6.
It must be -3.
On the inside, x is multiplied by something to give us 5.
It must be +5.
Similarly, we can factor the denominator using the AC method,
multiplying to -12 and adding to -1.
This is done with 4 and 3, if the 4 is negative.
Again, we will be careful with our factors.
To get 3x squared, we need 3x times x.
3x is multiplied by something to give us 3.
This is 3x times 1.
In the center, x times something must equal the other number, -4.
Now to factor the last fraction.
In the numerator, we're multiplying to AC, or -24,
and adding to 2.
This is done with 6 and 4, if the 4 is negative.
We get our 3x squared from 3x times x.
3x is multiplied by something to give us 6:
3x times +2 is 6.
In the center, x is multiplied by something to get -4.
It must be -4.
Similarly, in the denominator, when we multiply to AC,
we get -10 over 3.
This must be 5 and -2.
We build our last set of factors to get 2x is 2x times x.
Two had to be multiplied by something to get -2.
It must be -1.
In the center, x is multiplied by something to give 5.
It must be +5.
Finally, we have gone through
and factored each of these fractions
and can divide out any common factors
in the numerator and denominator.
The x plus 1s we find
in the numerator and denominator.
Same with the x plus 2.
We also find an x minus 1 and an x minus 3.
There is also a 2x plus 5 and a 3x minus 4.
Notice everything is divided out of these fractions.
This does not mean the answer is 0, but rather,
when everything divides out,
we're left with a 1 in the numerator
and a 1 in the denominator, and 1 over 1 will reduce to 1.
It is very important as we reduce,
we must factor all the polynomials first
to find common factors.