## Factor Theorem

The most factored form is the most factored version of a rational expression. Being able to find the most factored form is an essential skill when simplifying the derivatives found using product rule or quotient rule.

The most factored form is usually accomplished by common factoring the expression. But, any type of factoring may come into play.

##### How to Find the Most Factored Form

Set-Up: It’s easier to factor a rational expression if the coefficient is a fraction out front and if all functions are moved to the middle. Also, change all roots into rational exponent form. For example: $$\frac{3\sqrt{x^2+1}}{5(x-3)^2} \Rightarrow \frac{3}{5}(x^2+1)^{1/2}(x-3)^{-2}$$

Functions in Common: As with common factoring, any functions that are in common between terms should be taken out using the lowest exponent. For example: $$2x(x+3)^{-2}-3(x+3)^{-1}$$

The GCF would be $(x+3)^{-2}$

Coefficients in Common: For numerators, take out the greatest common factor between each numerator. For the denominators, take out the lowest common denominator. For example: $$\frac{4}{3}x(x+3)^{-2}+\frac{8}{9}(x+3)^{-1}$$

The coefficient of the GCF would be $\frac{4}{9}$

### Practice

EXAMPLE 1

Simplify the following to its most factored form: $$\frac{4x}{3(x+3)^2}+\frac{8}{9(x+3)}$$

###### SOLUTION

Using the Law of Division: $$2^{11} div 2^8=2^{11-8}=2^3$$

Evaluating: $2^3=8$

EXAMPLE 2

Simplify the following to its most factored form: $$\frac{\sqrt{x+2}}{\sqrt{x}}+\frac{\sqrt{x}}{\sqrt{x+2}}$$

###### SOLUTION

Using the Law of Powers: $$(m^0)^{10}=m^0$$

We should more properly use the Law of Zero Exponents to write: $m^0=1$

EXAMPLE 3

Simplify the following to its most factored form:

$$18x\sqrt{x+1}(2x-3)^{-1}+\frac{9}{2}x^2(x+1)^{-1/2}(2x-3)^{-1}-18x^2\sqrt{x+1}(2x-3)^{-2}$$
###### SOLUTION

Since we are multiplying the two powers, we use the Law of Multiplication: $$x+4=7$$

Therefore: $x=3$

EXAMPLE 4

Factor: $x^4-16$

###### SOLUTION

Since we are multiplying the two powers, we use the Law of Multiplication: $$x+4=7$$

Therefore: $x=3$