You can factor this trinomial by breaking it up into two binomials that lead with :

You will fill in the binomials by finding two factors of 36 that add up to 5. This is achieved with positive 9 and negative 4:

You can then set each of the two binomials equal to 0 and solve for :

The vertex form of a quadratic equation is given by

Where the vertex is located at

giving us .

Factor 2 as GCF from the first two terms giving us:

Now we complete the square by adding 4 to the expression inside the parenthesis and subtracting 8 ( because ) resulting in the following equation:

which is equal to

Hence the vertex is located at

When working with a rational expression, you want to first put your monomials in standard format.

Re-order the bottom expression, so it is now reads . 

Then factor a  out of the expression, giving you .

 The new fraction is  .

Divide out the like term, , leaving , or .

First factor the equation.  Find two numbers that multiply to 24 and sum to -10.  These numbers are -6 and -4:

Set both expressions equal to 0 and solve for x:

To find the greatest common factor of a polynomial expression, we need to find all of the factors each term has in common.  The easiest way to do this is to first look at the "coefficients", the "real numbers" to the left of the variable. 

The first term has "20," the second term has "30," and the third term has "10."

The biggest common factor of these three numbers is "10".

Next, we look at the variable, x. 

The first term has x^3, or three x's, the second term has x, or one x, and the third has x^2, or two x's.

The biggest common factor is "x", because while the first and third terms have more x's, the second term only has one x, so we can't pull out any more.

The greatest common factor is 10x, and when you divide each of the three terms by 10x, you get:

To find the GCF of the two terms, let's look at the coefficients, the numbers to the left of the variables, first.

The first term has a coefficient of 12, and the second has a coefficient of -24.  The GCF of 12 and -24 is 12, because that is the largest number that can divide BOTH 12 and -24 without leaving a remainder.

Next, let's look at the variables.  We have two types, "a" and "b".  The first term has one "a" and one "b" variable.  The second term has two "a's" and one "b" variable.

So, the most the terms have in common is ONE "a" and ONE "b"--that's the largest amount they have in common.

The GCF is 12ab, and when you divide the two terms by 12ab, you are left with:

Let's find the GCF of the coefficients first.  Remember, "coefficent" means the constant, the number to the left of the variables.

The first term has "27" and the second term has "-63" as their coefficients.  The GCF of 27 and -63 is 9.  9 is the largest number that can divide BOTH 27 and -63 without leaving a remainder.

Now, for the variables.  The first term has two "x" variables and one "y" variable.  The second term has three "x" variables and two "y" variables.  The greatest amount of x's and y's that the terms share is two x's and one y.

So, the GCF of the expression is .  When you divide the terms by this, you get:

Let's first look at the coeffiecients of these three terms.  They are: 25, 5, and -15.  The GCF of these three numbers is 5, because 5 is the largest number that can divide all of them without leaving a remainder.

So, "5" is the first part of our GCF.

Next, let's look at the variables.  We have two types, "a" and "b".  The first term has eight "b" variables, the second term has one "a" and one "b" variable, and the third term has two "a" variables.

THERE IS NO COMMON FACTOR FOR VARIABLES!

The first term has no "a" variables, and the third term has no "b" variables.  They have no variables in common.  So, we cannot factor out the variables. 

The GCF of the polynomial is simply "5".  When we divide each term by 5, we get:

Let's first find the GCF of the coefficients.  The coefficients of the four terms are -24, -4, 16, and 4.  The biggest number that can divide ALL of them without leaving a remainder, is 4.

The first part of our GCF is "4."

Next, let's look at the variables.  We have only one type, "x."  The first term has four "x" variables, the second has two, the third has eight, and the fourth has six.

The greatest amount of "x" variables that each term has in common is 2.  Each has at least two x variables.  The 1st, 3rd, and 4th terms have MORE than two x variables, but since the 2nd term only has two x's, that's the greatest amount that we can factor from ALL of the terms.

So, our GCF is 4x^2.  When we divide each of the terms by 4x^2, we get: