**Three Types of Factoring**

There are three basic types of factoring:

1. Factoring out a common monomial:

18 x^{2}– 9x= 9x(2x– 1)ab–cb=b(a–c)

2. Factoring a quadratic trinomial using FOIL in reverse

*x*^{2} – *x* – 20 = (*x* – 5)(*x* + 4)

*x*^{2} – 6*x* + 9 = (*x* – 3)(*x* – 3) = (*x* – 3)^{2}

(for more on quadratic trinomials, see Quadratic Trinomials, Quadratic Equations, and the Quadratic Formula Help):

3. Factoring the difference between two perfect squares using the rule *a*^{2} – *b*^{2} = (*a* + *b*)(*a* – *b*):

*x*^{2} – 81 = (*x* + 9)(*x* – 9)

*x*^{2} – 49 = (*x* + 7)(*x* – 7)

**Using Common Factors**

With some polynomials, you can determine a common factor for all of the terms. For example, 4x is a common factor of all three terms in the polynomial 16*x*^{4} + 8^{2} + 24*x* because it can divide evenly into each of them. To factor a polynomial with terms that have common factors, you can divide the polynomial by the known factor to determine the second factor.

**Example**

In the binomial 64*x*^{3} + 24*x*, 8*x* is the greatest common factor of Both terms. Therefore, you can divide 64*x*^{3} + 24*x* by 8*x* to find the Other factor.

= 8*x*^{2} + 3

Thus, factoring 64*x*^{3} + 24*x* results in 8*x*(8*x*^{2} + 3).

**Fractions, Variables, and Factoring**

**Isolating Variables Using Fractions**

It may be necessary to use factoring in order to isolate a variable in an equation.

**Example**

If a*x* – *c* = b*x* + *d*, what is x in terms of *a*, *b*, *c*, and *d*?

First isolate the *x* terms on the same side of the equation:

*ax* – *bx* = *c* + *d*

Now factor out the common *x* term:

*x*(*a* – *b*) = *c* + *d*

Then divide both sides by *a* – *b* to isolate the variable *x*:

Simplify:

*x* =

**Fractions with Variables**

You can work with fractions that contain variables in the same way as you would work with fractions without variables.

**Example**

Write as a single fraction.

First determine the LCD of 6 and 12: The LCD is 12. Then convert each fraction into an equivalent fraction with 12 as the denominator:

Then simplify:

**Reciprocal Rules**

There are special rules for the sum and difference of reciprocals. The following formulas can be memorized to save time when answering questions about reciprocals:

- If
*x*and*y*are not 0, then +*y*=*x*+ - If
*x*and*y*are not 0, then =*y*–

**Finding Roots**

The roots of an equation are the values that make the equation true. For example, what are the roots of *x*^{3} – 9*x*^{2} – 10*x* = 0?

First, factor out the variable *x*: *x*(*x*^{2} – 9*x* – 10). Then, factor again: *x*^{2} – 9*x* – 10 = (*x* – 10)(*x* + 1), and *x*^{3} – 9*x*^{2} – 10*x* = *x*(*x* – 10)(*x* + 1). Set each factor equal to 0 and solve for *x*: *x* = 0; *x* – 10 = 0, *x* = 10; *x* + 1 = 0, *x* = –1. The roots of this equation are 0, –1, and 10.

**Undefined Expressions**

A fraction is undefined when its denominator is equal to 0. If the denominator of a fraction is a polynomial, factor it and set the factors equal to 0. The values that make the polynomial equal to 0 are the values that make the fraction undefined.

For what values of *x* is the fraction undefined?

Factor the denominator and set each factor equal to 0 to find the values of *x* that make the fraction undefined: *x*^{3} + 7*x*^{2} = *x*^{2}(*x* + 7), *x*^{2} = 0, *x* = 0; *x* + 7 = 0, *x* = –7. The fraction is undefined when *x* equals 0 or –7.

**Radicals**

In order to simplify some expressions and solve some equations, you will need to find the square or cube root of a number or variable. The **radical** symbol, √ , signifies the root of a value. The square root, or second root, of *x* is equal to √*x*, or ^{2}√*x*. If there is no root number given, it is assumed that the radical symbol represents the square root of the number. The number under the radical symbol is called the **radicand**.

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