The most factored form is usually accomplished by common factoring the expression. But, any type of factoring may come into play.
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1 MOST FACTORED FORM 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: 3 x # x 3 # 5 x# + 1 (/# x 3 *# 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: è The GCF would be x + 3 *# 2x x + 3 *# 3 x + 3 *( 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: 4 3 x x + 3 *# x + 3 *( è The coefficient of the GCF would be: 4/9. EXAMPLE: Simplify the following to its most factored form: a) 4x 3 x + 3 # + 8 9(x + 3) = 4 3 x x + 3 *# x + 3 *( = 4 9 x + 3 *# [3x + 2(x + 3)] 14 Factoring Most Factored Form
2 Then, simplify within the bracket: = 4 9 x + 3 *# (5x + 6) 4(5x + 6) = 9 x + 3 # b) x + 2 x x x + 2 = x *(/# x + 2 (/# x (/# x + 2 *(/# = x *(/# x + 2 *(/# [ x + 2 x] Then, simplify within the bracket: = 2x *(/# x + 2 *(/# 2 = x(x + 2) c) 18x x + 1 2x 3 *( x# x + 1 *(/# 2x 3 *( 18x # x + 1 2x 3 *# = 18x x + 1 (/# 2x 3 *( x# x + 1 *(/# 2x 3 *( 18x # x + 1 (/# 2x 3 *# = 9 2 x x + 1 *(/# 2x 3 *# 4(x + 1)(2x 3) + x(2x 3) 4x(x + 1) Then, simplify and factor within the bracket: = 9 2 x x + 1 *( # 2x 3 *# 6x # 11x 12 = 9x 6x# 11x 12 2 x + 1 2x 3 # 15 Factoring Most Factored Form
3 PRODUCT RULE Suppose that the function you must differentiate can be written as the product of two functions, f(x) and g(x): y = f x g x Let f ) x and g (x) be the derivative of f(x) and g(x). The derivative of y can be found by taking: The derivative of the first function times the second PLUS the derivative of the second function times the first. In another way: y = f x g x y ) = f ) x g x + g (x) f(x) EXAMPLE: Find the derivative of the function: y = 3x. + 2x 1 x. + 5x + 4 SOLUTION: We can see that this can be expressed as the product of two functions: f x = 3x. + 2x 1 and g x = x. + 5x + 4 The derivative of each function individually is: f ) x = 6x + 2 = 2(3x + 1) and g x = 2x + 5 So the derivative of the original function y is: y ) = f ) x g x + g ) x f x y = 2 3x + 1 x. + 5x x + 5 3x. + 2x 1 Now factor to achieve the most factored form: y ) = 2 3x + 1 (x + 1)(x + 4) + 2x + 5 3x 1 x + 1 y ) = x x + 1 x x + 5 3x 1 y ) = (x + 1)(6x. + 39x + 13) Avoiding Product Rule Product rule is a longer and more difficult process that can sometimes be avoided in order to make less mistakes. If the original polynomial function is easy to expand, do that first, then find the derivative. For example: f x = 3x 1 2x + 1 = 6x. + x 1 f ) x = 12x Product Rule
4 QUOTIENT RULE Suppose that the function you must differentiate can be written as the quotient of two functions, f(x) and g(x): f x y = g x Let f ) x and g (x) be the derivative of f(x) and g(x). The derivative of y can be found by taking: The derivative of the top function times the bottom MINUS the derivative of the bottom function times the top ALL OVER the bottom squared. In another way: y = f x g x y ) = f) x g x g (x) f(x) g x. EXAMPLE: Find the derivative of the function: y = 2x 1 x. + 1 SOLUTION: We can see that this can be expressed as the quotient of two functions: f x = 2x 1 and g x = x. + 1 The derivative of each function individually is: f ) x = 2 and g x = 2x So the derivative of the original function y is: y ) = f) x g x g ) x f x g x. y = 2 x x(2x 1) We can t factor the numerator so we ll expand and simplify: y ) = 2x x. + 2x = 2x. + 2x + 2 y = 2(x. x 1) 6 Quotient Rule
5 Avoiding Quotient Rule Quotient rule is a longer and more difficult process that can sometimes be avoided in order to make less mistakes. If the denominator is a monomial, split the denominator over each term in the numerator. For example: f x = x. 2x + x. Now, we can use power rule: x = x. x. 2x x. + x x. = 1 2x67 + x 68/. Simplify to its most factored form: f ) x = 2x x6:/. f ) x = 1 2 x6:/. 2x 7/. 3 f ) x = 2 x 3 2 x : 7 Product Rule
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