4

Factoring

Factoring is done with quadratic polynomials. A quadratic polynomial is a polynomial with the power of 2. Any quadratic equation can be factored simply or if not can be done with the Quadratic Equation (see below). Factoring is another word to simplify the polynomial. For now let's take a look how to factor a quadratic polynomial. The solution to the equation is two x values. They can be the same x value or two different x values. The x values are called roots. We use those roots when we want to graph the equation.


Example:

  1. x2 + 4x + 4 = 0

    2. We first notice that the first term is x2, so we know that we are going to factor. It takes the form of x2 + 4x + 4 = (x + __)(x + __).

    It takes this form because the only way to get the first term is to multiply x by x. Therefore, the first term in each factor must be an x. To complete this all we need to do is find the two numbers that will go into the blank spots.

    To find the missing numbers we need to figure out what two numbers when multiplied together get 4, which means that they must be factors of 4. Those two numbers also when added need to be equal to 4. We can start by listing all the possible ways to get to 4 by multiplication. Here are all the possibilities: (1)(4) and (2)(2) Since we are working with only positive numbers we don't have to worry about negative numbers.

    From those choices we can tell that there is only one option: (2)(2).

    Plugging the numbers into the equation and expanding we get x2 + 4x + 4 = (x + 2)(x + 2), which what we wanted.

In this example the roots would be x = -2. There is only one root since the x value is the same. To find the roots we set each factor to 0. In this case, we have x + 2 = 0. Solving for x we get that x = -2.

FOIL

To find the solution from the simplified form is called expanding the quadratic. Another way to say expanding the quadratic is to FOIL. FOIL stands for First Outter Inner Last.

Example

  1. (x + 4)(x - 3)
    2. First: (x)(x) = x2
    Outter:(-3)(x) = -3x
    Inner:(4)(x) = 4x
    Last: (4)(-3) = -12
    Putting it together we get x2 - 3x + 4x - 12. Combining like terms -3x + 4x we get x. Finally we get our solution: x2 + x - 12.

In this example we would have two roots, x = -4 and x = 3. Once again setting each factor equal to 0. We would get x + 4 = 0 and x - 3 = 0. Solving both equations we end up with our roots -4 and 3.


Quadratic Equation

Another way to solve a quadratic polynomial is to use the quadratic equation. A quadratic equation is in the form of ax2 + bx + c = 0. We use the quadratic equation when we can't factor the polynomial easily. Sometimes we use it even when the polynomial can easily be factored or to check our answers. We use factoring to find the roots when we graph the equation and the quadratic equation is the perfect way to get those roots.

The formula to solve a quadratic equation is

x =
-b ±√b2 - 4ac
/ 2a

Example

Use the quadratic equation to solve the following.

  1. x2 + 2x - 3 = 0

    2. The first thing to figure out is which number correspondes to which letter.
    a = 1
    b = 2
    c = (-3)

    Now we plug them into the equation.

    x =
    -(2) ±√(-2)2 - 4(1)(-3)
    / 2(1)

    x =
    -2 ±√4 + 12
    / 2(1)

    x =
    -2 ±√16
    / 2

    x =
    -2 ±4
    / 2

    Now we break up the fraction into two separate equations, one for the plus and one for the minus.


    Plus Minus
    x =
    -2 + 4 / 2

    x =
    2 / 2

    x = 1     		x =
    -2 - 4 / 2

    x =
    -6 / 2

    x = -3

    After doing all of these we end up with two solutions x = 1 and x = -3. If we were to factor this out we would have the same solutions.