# Algebra 1 : How to factor a polynomial

## Example Questions

### Example Question #31 : How To Factor A Polynomial

Factor

Explanation:

To factor this polynomial, first think of the pairs of numbers that when multiplied, equal

and  are a good place to start.

Then, you know that one of them has to be negative since your last term in the polynomial is negative. Try different combinations to get a . This would mean that you have a  and .

### Example Question #31 : How To Factor A Polynomial

Factor this polynomial:

Explanation:

First, a  term can be factored out to give a simplified polynomial:

Then, the polynomial can be factored to give the correct answer:

### Example Question #1 : Quadratic Equations

Factor:

Explanation:

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:

### Example Question #1 : Quadratic Equations

Solve for :

Explanation:

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 :

### Example Question #1 : Parabolic Functions

Write a quadratic equation having  as the vertex (vertex form of a quadratic equation).

Explanation:

The vertex form of a quadratic equation is given by

Where the vertex is located at

giving us .

### Example Question #1 : Graphing Parabolas

Find the vertex form of the following quadratic equation:

Explanation:

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

### Example Question #31 : How To Factor A Polynomial

Simplify:

Explanation:

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 .

### Example Question #1 : Finding Roots

Solve for .

Explanation:

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:

### Example Question #32 : How To Factor A Polynomial

Find the greatest common factor (GCF) of the following polynomial expression.

Explanation:

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:

### Example Question #32 : How To Factor A Polynomial

Find the greatest common factor (GCF) of the following polynomial expression.

Explanation:

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: