Turning Standard Form to Factored Form

Factoring to turn to Factored Form

What is factoring?

Factoring is the process of finding factors of an expression. Factors are simply two numbers that multiply to give a product.

Quadratic relations like x^2 - 8x + 36 can also be factored.

(x - 6) (x-6) = x^2 - 8x + 36

FACTORS

Note how the expression on the right is in factored form, while the expression on the right is in standard form. When factored form is expanded, the answer will be in standard form.

While factored form can be turned to standard form, standard form can also be converted to factord form using these methods:

Greatest Common Factor
Simple Trinomials
Complex Trinomials
Perfect Squares
Difference of Squares

Greatest Common Factor

When factoring using the greatest common factor, you must look for what is the most common between all the terms of the expression, including the variables.

EXAMPLE 1

Find the Gratest Common Factor of the expression:

y = -5x^2 - 10x - 20

In this quadratic relation, all the terms can be divided by 5, and also, a negative can be taken out as all the terms are negative.

y = -5x^2 - 10x - 20

-5 -5 -5

y = -5 (x^2 + 2x +4)

Since nothing else is common between the terms of the quadratic equation now, another factor cannot be taken out, so the final answer is y = -5(x^2 + 2x +4).

To check if your answer is right, convert this factored form into standard form and see if your answer is y = -5x^2 -10x -20.

EXAMPLE 2

Find the Gratest Common Factor of the expression:

y = -6x^4 + 12x^3 - 24x^2

In this quadratic relation, all the terms can be divided by 6, but not -6 because the term 12x^3 is not negative. However, all the terms in this equation do have the variable x, hence all the terms can be divided by 6x^2, as x to the power of 2 is the lowest power of x.

y = -6x^4 + 12x^3 - 24x^2

6x^2 6x^2 6x^2

y = 6x^2 (-x^2 + 2x - 4)

Since nothing else is common between the terms of the quadratic equation, another factor cannot be taken out, so the final answer is y = 6x^2 (-x^2 + 2x - 4).

To check if your answer is right, convert this factored form into standard form and see if your answer is y = -6x^4 + 12x^3 - 24x^2.

A simple trinomial is a polynomial or an expression with 3 terms. In a quadratic expression with 3 terms, if there is a coefficient in front of the x^2 or in other words, if the value of a is greater than 1, it is simple trinomial.

When factoring, we try to find 2 expressions that will multiply to give a product of the original equation. To factor simple trinomials, we first work with the term x^2. There is only one method of factoring this term, as the only factors of x^2 are x and x. This is why the factored form of a quadratic will always be in from "(x + n)(x + m)". This is why the parentheses will always have the variable 'x' in the start.

(x )(x )

Now, to find the missing values, we need to determine what 2 numbers will multiply to the c value of the standard form equation, and add to the b value. The answer will be you m and n values.

Simple Trinomials

EXAMPLE

y = x^2 + 5x - 24

To factor this equation, put all the value in a chart like this:

x^2 -24

A chart like this makes it easier to visualize the equation and also to find the factors. What this chart shows is that we need to find 2 numbers that multiply to -24 and add to 5.

In this column, we need to find two variables that multiply to x^2, and those variables are x and x.

x^2 -24

-3

The 2 numbers that multiply to -24 and add to 5 are 8 and -3. Therefore, the factored form of this equation is:

8 x -3 = -24

8 + (-3) = 5

(x + 8) (x - 3)