## 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:** Itâ€™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

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

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

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

Factor: x^4-16

###### SOLUTION

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

Therefore: x=3

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