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:

We are going to use factoring by grouping to factor this expression.

The expression:

has the form:

In factoring by grouping, we want to split that B value into two smaller values a and b so we end up with this expression:

The rules are:

and

For our problem A=6 B=-13 C=6

So we have:

Another way of saying this is that we are looking for factors of 36 that add up to -13

The values that work are:

Plugging those values in:

Now let's group and factor out from both groups:

Finally let's factor out a (2x-3) from both terms: and we get our answer:

The key to factoring the polynomial is recognizing a "product of a sum and a difference". The resulting inner terms cancel each other out in the final expanded polynomial.

.

One could use the FOIL and grid methods to factor the polynomial into

.

Leaving as the correct answer.

Here . Multiply and and you get which can be factored as and and when one adds and you get . Hence the quadratic equation can be rewritten as

Now you factor by grouping the first two terms and the last two terms giving us

which can be further factored resulting in

By setting each of the two factors to 0 we get

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:

When asked to factor a difference of squares, the solution will always be the square roots of the coefficients with opposite signs in each pair of parentheses.

For the quadeatic equation . Applying these to the quadratic formula

we get

resulting in

and

Begin by factoring out a 2:

Then, we recognize that the trinomial can be factored into two terms, each beginning with :

Since the last term is negative, the signs of the two terms are going to be opposite (i.e. one positive and one negative):

Finally, we need two numbers whose product is negative thirty-five and whose sum is positive two. The numbers and fit this description. So, the factored trinomial is:

Because both terms are perfect squares, this is a difference of squares:

The difference of squares formula is .

Here, a = x and b = 5. Therefore the answer is .

You can double check the answer using the FOIL method:

The product of the last two numbers should be 6, while the sum of the products of the inner and outer numbers should be 5x. Factors of six include 1 and 6, and 2 and 3. In this case, our sum is five so the correct choices are 2 and 3. Then, our factored expression is (x + 2)(x + 3). You can check your answer by using FOIL.

y = x2 + 5x + 6

2 * 3 = 6 and 2 + 3 = 5

(x + 2)(x + 3) = x2 + 5x + 6