needs to be seen as . Then we need to ask what factors of "C" will equal "B" if one of them is multiplied by "A"

So first thing is to find factors of "C," luckily 5 only has 2:

1 and 5

We need to write an equation that uses either 5, 1 and a "*2" to equal -9.

 Now we can write out our factors knowing that we will be using -5,2,and 1.

(__X+___)(___X+____)

from the work above, we know we have to multiply the 2 and the -5, so they need to be in opposite factors

(2X+__)(X-5)

that only leaves one space for the 1

needs to be seen as . Then you need to ask what factors of "C" will equal "B" when added together.

 C=-18, factors of -18 are:

-1,18

-2,9

-3,6

1,-18

2,-9

3,-6

Of those factors, only -3,6 will give us "B", which in this case, is "3"

so

becomes

Set , and, consequently, . Substitute to form a quadratic polynomial in :

Factor this trinomial by finding two numbers whose product is 8 and whose sum is . Through trial and error, these numbers can be found to be and , so

Substitute  back for :

The first binomial is the difference of squares; the second is prime since 8 is not a perfect square. Thus, the final factorization is

Set , and, consequently, . Substitute to form a quadratic polynomial in :

Factor this trinomial by finding two numbers whose product is 8 and whose sum is . Through trial and error, these numbers can be found to be and , so

Substitute  back for :

Both factors are the difference of perfect cubes and can be factored further as such using the appropriate pattern:

Step 1: Break down  into factors...

Step 2: Find two numbers that add or subtract to .

We will choose  and .

Step 3: Look at the equation and see which number needs to change sign..

According to the middle term,  must be negative.

So, the factors are  and .

Step 4: Factor by re-writing the solutions in the form:

So...

Rationalize the following fraction:

To rationalize a denominator, we will multiply the top and bottom of the fraction by the denominator.

And we have our answer

First we need to factor both polynomials.

Becomes

Now we cancel out any variables that are in BOTH the numerator and denominator. Remember that if a group of variables/numbers are inside parenthesis, they are considered a single term.

The common term in this case is , removing that from the equation gives us

The first step is to factor both polynomials:

  Becomes

Now we cancel out any terms that are in both the numerator and denominator. Remember that any variables/numbers that are in parenthesis are considered a single term.

In this case, the common term is

Once we remove that, we are left with:

The first step is to factor both polynomials

   becomes

Now we cancel out any terms that are in both the numerator and denominator. Remember that any variables/numbers that are in parenthesis are considered a single term.

In this case, the common term is

Simplified, the equation is:

The first step is to factor both polynomials

becomes

Now we cancel out any terms that are in both the numerator and denominator. Remember that any variables/numbers that are in parenthesis are considered a single term.

In this case, the common term is

Simplified, the equation is: