MATH 0322 Intermediate Algebra Unit 1 GCF Section: 6.1 GCF Terminology: factoring : the process of writing a polynomial as a product of its multipliers(factors) GCF : Greatest Common Factor - an expression of the highest degree that divides each term of a polynomial. polynomial : a single term or the sum of 2 or more terms containing variables with whole number exponents. *Important*- when factoring polynomials, the variable part of its GCF always contains the smallest power of a variable that appears in all of its terms. GCF Practice: Find the GCF for and .

1. Factor each term: 3 2 12 =2 2 3 1 8 2 =2 3 3 2. Circle common factors. 3. From either row, multiply GCF factors to simplify. GCF 2 3 =6 2 P GCF To factor a monomial from a polynomial: 1. Factor each term and determine GCF 2. Write GCF( ) 3. In ( ), simplify each remaining term to complete the factorization.

Practice: Factor 2 2 3 2 2 5 +2 2 2 GCF GCF 4 ( 3 5 +2) P 4 ( 3 2 5 +2) GCF GCF Factoring when the leading coefficient is negative: 1. Factor each term and determine GCF 2. Write GCF( ) 3. In ( ), simplify each remaining term, then change signs to complete

the factorization. Practice: Factor 2 2 3 + 2 2 5 2 2 2 4 (3 +5 2) 4 (3 2+ 5 2) P 4 (3 2 5 + 2) GCF Polynomials having four terms Normally, the four terms only have a common factor of 1. Rearranging the four terms will result in factorable pairs. By grouping these factorable pairs, the polynomial can then be factored. First, identify the common factor to complete the

factorization. ( +6)( 2 +3) P ( +2)( 7) P GCF How to factor a polynomial having four terms 1. Rearrange(group) terms having common factor, if needed. 2. Factor out monomial GCF from each pair of terms. 3. Identify and factor out binomial GCF, if one exists. Practice: Factor P MATH 0322 Intermediate Algebra Unit 1 Factoring Quadratic Trinomials Section: 6.2-6.3

Factoring Quadratics Multiplication facts are essential to factoring. Practice: As quickly as you can, list all of the integer products for the number below. Factoring Quadratics GCFMethod Your first Factoring Method is: _____ Your next Factoring Method will rely on your understanding of the FOIL Method for binomial multiplication. Once this method is mastered, you will add it and the GCF Method to your Factoring Strategy on polynomials. In this section, you will factor trinomials having a leading coefficient of 1. **Careful** some trinomials are considered prime(not factorable). Factoring Quadratics

FOIL Method : a method used to multiply two binomials. Example: Multiply using the FOIL Method. Simplify F O I L First term First term Outside term Outside term Inside term Inside term Last term Last term ** You will use the understanding of multiplying binomials this way to help factor . P Factoring Quadratics

Remember, you are learning how to rewrite an expression as a product of its factors. Factored Form Trinomial Form Practice: If , then use this to backtrack the steps that got P 2 +6 +8 What factors Whathere, factors here, multiply & multiply & produce produce

but also add & produce Factoring Quadratics Practice: Factor P First Factors Last Factors are Which factors add and give sum of ? Check answer by multiplying binomial factors. Factoring Quadratics Practice: Factor

P First Factors Last Factors are Which factors subtract and give difference of ? Check answer by multiplying binomial factors. Factoring Quadratics Practice: Factor P Common Factors First Factors Last Factors are

Which factors add and give sum of ? (both factors must be negative) Check answer by multiplying all factors. MATH 0322 Intermediate Algebra Unit 1 Factoring Special Forms Section: 6.4 Factoring Special Forms This section presents methods for factoring Special Forms of algebraic expressions. In factoring these expressions, it is helpful to learn to recognize their special characteristics. (Hint: Look for Perfect Squares and Cubes.) In the examples given below, state any special characteristics that you see.

() () Perfect Squares () ( ) () Perfect Squares () ()

Perfect Cubes Factoring Special Forms Study the first example more closely and begin to visualize the FOIL Method used to factor in the previous section. 2 ___ +0 16 _____ _____ First First Last Last Outside Inside Difference

of Squares ( ) ) 4 ( 4 + P 4 +4 0 Factoring Special Forms Factor using Difference of Squares: 1)

(+ ) ) 6 ( 6 2) ( ) ) 1 ( 1 3) 5 ) 5( + ) 7 ( 7

+ P P P Factoring Special Forms Study the second example more closely on the first slide. Try to visualize the FOIL Method to factor it as in the previous section. middle sign is positive ___ _____ _____ First First Last Last Outside Inside

Perfect Square Trinomials + P 3 ( ) ( ) 3 3 +3 + 6 Factoring Special Forms Factor using Perfect Square Trinomials:

middle sign is positive + P ( ) ) 5 ( 5 + 1) middle sign is negative middle sign is positive 3) P (+ )

2 +) 2 3 ( 3 P ( ) ) 8 ( 8 2) Factoring Special Forms Study the third example from the first page more closely and begin to visualize how the Cube Formulas are used to factor it. 2 ( ( +2 )

+4 2 ) 3 3 () +(2) Sum and Difference of Cubes P Factoring Special Forms Factor using Sum or Difference of Cubes. ( ) 2 4

( +4 ) +16 3 3 () +(4 ) 2 + 2

P MATH 0322 Intermediate Algebra Unit 1 General Factoring Strategy Section: 6.5 General Factoring Strategy When factoring polynomials, follow the General Factoring Strategy below as discussed in class: I. ______ GCF Method II. _________ SpecialProducts ( +)( )

( + )( + ) ( )( ) ( + )( 2 + 2 ) ( )( 2+ + 2 ) III. _______ & _______) Error FOIL Method (or ______ Trial IV. ___________ GroupingMethod (for polynomials having four unlike terms) **Careful** some trinomials are prime. General Factoring Strategy Practice: Factor I. II. III. IV.

P GCF? Special Product? FOIL Factor? Grouping? P P Check answer P 2 5 () no no

General Factoring Strategy Practice: Factor I. II. III. IV. P GCF? Special Product? FOIL Factor? Grouping? P P Check answer P

no no General Factoring Strategy Practice: Factor P I. GCF? II. Special Product? P P P III. FOIL Factor? IV. Grouping? Check answer

no no General Factoring Strategy Practice: Factor I. II. III. IV. P GCF? Special Product? FOIL Factor? Grouping? P P Check answer

no no no P 4 4 2 +9 General Factoring Strategy Practice: Factor P I. GCF? II. Special Product?

P P P III. FOIL Factor? IV. Grouping? Check answer no no MATH 0322 Intermediate Algebra Unit 2 Rational Expressions Section: 7.1 Rational Expressions quotientof two

Rational Expressionis the ________ A __________________ polynomials ___________. , , , Examples are: The denominator of a rational expression can never be allowed to be ____. zero division A rational expression indicates _______, undefined and division by zero is _________. exclude any value(s) Always ________________of the variable that undefined cause a rational expression to be __________. Rational Expressions Quick Check: For what value(s) is the rational

expression undefined? undefined when Solve for . =4 is undefined when . undefined when Factor to Solve for . P (+5 )()=0 5

+5=0 5=0 =5 =5 is undefined when . P Rational Expressions fraction A rational expression is like a _______. It can be reduced if the numerator and denominator have factors common ______. Fundamental Principle of Rational Expressions 0 polynomials If , , and are __________ with and ,

= then To simplify a rational expression requires: Factoring 1) ________ numerator and denominator completely, reducing 2) then ________ any common factors Some rational expressions cant be simplified. 0 Rational Expressions Practice: Simplify if possible. 1) Factor numerator and denominator. 2) Reduce common

factors. 3 2 4 1 1 P Rational Expressions Practice: Simplify if possible. 1) Factor numerator and denominator. 2) Reduce common factors. 3 +2 5 2 No common factors. The expression cannot be simplified.

P Rational Expressions Practice: Simplify if possible. 1) Factor numerator and denominator. 2) Reduce common factors. +3 +6 +1 +3 1 1 P MATH 0322 Intermediate Algebra Unit 2 Multiplying and Dividing

Rational Expressions Section: 7.2 Multiplying and Dividing Multiplying rational expressions is just like multiplying fractions. To multiply , you would : 1) multiply numerators across 756 36 21 = 18 45 18 45 2) multiply denominators across 756 756 = 810 18 45

3) then reduce common factors ! ,h ! 756 = 810 Multiplying and Dividing Instead, to make life easier, we have learned to do this in reverse by: 36 21 2 2 3 3 3 7 1) first factoring completely 18 45 =2 3 3 3 3 5 1 1 1 1 2 2 3 3 3 7 2) reducing common factors 2 3 3 3 3 5 1 1 1 1

3) then multiplying across P 1 2 1 1 1 7 14 = 1 1 1 1 3 5 15 hhhhh ! , h ! Multiplying and Dividing Principle for Multiplying Rational Expressions If , , , and are rational expressions, then . To multiply rational expressions, follow this same strategy: 1) first factoring completely

2) reduce common factors 3) then multiply across Multiplying and Dividing Principle for Dividing Rational Expressions If , , , and are rational expressions, where , , and , then . To divide rational expressions, you need to: 1) first change division to multiplication 2) change the second value to its reciprocal 3) then follow the multiplication strategy. Multiplying and Dividing Quick Check: When multiplying or dividing rational expressions, what do you do if an expression appears without a denominator?

1 1 .then continue as usual. Practice Multiply as indicated: Factor. 1) 36 (4 )(9 (4)(9) = 9 +39 91 +3 +3

Reduce. (common factors) Simplify. 4 +3 P Practice Your Turn. Multiply as indicated: 1 Factor. 56 (8 )( 7 ) =

9 8 9 8 1 Reduce. (common factors) Simplify. 7 9 P Practice Multiply as indicated: Factor. 1

1 10 +25 9 9 10 +25 9 5(2 +5) = 2 2 +5 9) 2 +5 9 2 +5 ( 9 2 1 1

Reduce. (common factors) Simplify. 5 P Practice Your Turn. Multiply as indicated: Factor. 1 1 6 +8 7 10 +25

7 9 2(3 +4 ) = 2 2 +5 7 3+4 3 +4 ( 97) 2 1 1

Reduce. (common factors) Simplify. 2 P Practice as indicated: Multiply Special Product Factor. 1( 2 9) 2+ 14 + 48 2 2

+5 +6 +5 24 1 1 1 1 0 +3)( +25 9 1( 3) ( +6)( +8) 2 +5 ( +

3)(9 + 2) ( 2 3)( +8) 1 1 1 Reduce. (common factors) Simplify. +6 +2 P Practice

Your Turn. Multiply as indicated: 2 2 +12 +27 1( 4 ) 2 2 +11 +18 +3 10 Factor. 1 1 1 1 03)(+ +25 2) 9 (+ 9) 1(

+2)( 2 2 +5 ( + +9) (+5)( 2) 2)( 9 1 1 1 Reduce. (common factors) Simplify. + 3 +5

P Practice Divide as indicated: 2 25 4 8 5 Factor. 1 10 52 (5)(5) 9 2 42(+5

2) 5 9 1 1 1 Reduce. (common factors) Simplify. 5 4 P Practice Your Turn. Divide as indicated: + 2 72

8 5 +10 Factor. 1 10 +52 (8)(9) 9 2 2 +5 5( +2) 8 9 1

1 1 Reduce. (common factors) Simplify. 9 5 P Practice Divide as indicated: 4 ( 2 36) 5( 2 +4 12) 4 2 144 5 2 +20 60 2 2

+12+36 8 + 12 Factor. Reduce. (common factors) Simplify. 1 1 1 1 1 0 5 9 4 ( +6)( 6)

5( +6)( 2) +6 6 +6 2 2 2 )( +5 (6 2) 9 +6 (+ 4)(

+8) +6 6 2 1 1 1 1 20 1 20 P End Section 7.2 MATH 0322 Intermediate Algebra Unit 2 Adding and Subtracting Rational Expressions

(Unlike Denominators) Section: 7.4 Adding & Subtracting Here is the Principle of Adding and Subtracting Rational Expressions stated in this chapter. If and are rational expressions, then and . Adding & Subtracting Adding & Subtracting rational expressions follows the same rules as for fractions. To Add , you would : 3 1 + 8 8

1) add the numerators across, (since denominators are the same) common denominator, 2) keep the ________ 3 +1 8 1 3) then reduce common factors. (To reduce rational expressions, factoring will be needed.) P

4 1 8 2 2 Adding & Subtracting ..but what do you remember to do whenever the denominators are unlike? To Add , you would : find a _____ ________ Denominator Common for andLeast , build up each fraction with the LCD, then add as usual.

3 2 3 1 + 8 12 P 11 9 2 + = 24 24 24 Adding & Subtracting finding an LCD for polynomial denominators requires factoring. To find the LCD for and

1) factor each denominator polynomial P P P P 3 LCD 48 2 12 =2 2 3 2 2 2cross 2 out matching factors, 1 6 =2 2#2: 2) Polynomial

3 multiply remaining factors to Polynomial #1 3) multiply Polynomial #1 to complete LCD. P Adding & Subtracting Practice: Find the LCD for 1. 3. 2. and . PP P PPP 24 = 2 2 2 3 3 5 3 6 3=2 2 3 3

LCD P 72 5 Adding & Subtracting Practice: Find the LCD for and P +2 ( )( 2)( 3) LCD 4 = ( 4 ) = ( )( ) 3 +2

2 6= 3 2 LCD is or leave LCD in Pbut factored form for later. Adding & Subtracting Practice: Add. Factor and Find LCD. 2 6 + +5 3

LCD (+5)( 3) Build up expressions ( )2 6(+ ) + with LCD. ( )( + 5) ( 3)(+ ) 26 6 + 30 + ( + 5)( 3) ( + 5)( 3) Add as usual. (2 6 )+(6 +30) ( + 5)( 3) 8 ( + 3) 8 + 24 =

( + 5)( 3) ( + 5)( 3) P Adding & Subtracting Practice: Subtract. Factor and Find LCD. LCD ( 5)(5) Build up expressions with LCD. Subtract as usual. 5 2 5 5 25 ( 5) 5( 5)

5 ( 5) 5 ( 5) 25 2 ( 5) (5) ( 5)(5) 1 ( 2 25) ( +5)( 5) +5 25 2 = = = 5

( 5)(5) ( 5)(5) ( 5)(5) 1 P Adding & Subtracting So, Adding and Subtracting rational expressions is just like it is for fractions, to Add or Subtract, the rational expressions must have like or ________ common denominators. to Add or Subtract rational expressions having unlike denominators, 1) first find an ____ LCD for the denominators and _____ build up each rational expression, 2) add or subtract as usual, 3) then _______ reduce by _________.

factoring MATH 0322 Intermediate Algebra Unit 2 Complex Rational Expressions Section: 7.5 Complex Rational Expressions Complex Rational Expressions are rational expressions in which the numerator and denominator can each contain one or more rational expressions. Examples: , , Complex rational expressions are also called complex _________. fractions Two methods will be outlined for simplifying complex rational expressions: 1) ________________ Division Method

Form a single rational expression for the numerator and denominator, then divide. LCD Method 2) _____________ Eliminate all denominators with an LCD, then simplify. Complex Rational Expressions A rational expression is considered simplified when it is in the form , where and are polynomials with no common factors. Examples: , , Practice, practice, practice. These skills will help prepare you for the remainder of this course and are required in College Algebra. Complex Rational Expressions Division Method: Simplify by dividing Simplify:

1) Combine the numerator: 2) Combine the denominator: 3) Divide, then reduce: 1 2 11 3 + = + 8= 4 3 12 12 12 2 1 5 8 = 3= 3 4 12 12 12 11

11 12 = 5 12 12 1 12 1 11 5 5 P Complex Rational Expressions Division Method: Simplify by dividing Simplify:

1) Combine the numerator: 2 1 = 2 1 2 1 = 1 2) Combine the denominator: 3) Divide, then reduce: 2 + 1 =2 + 1=2+1 1 2 1 2 1 1 =

2 + 1 2 +1 1 21 2 +1 P Complex Rational Expressions Division Method: Simplify by dividing Simplify: 1) Combine the numerator: 1 1

= = 2) Combine the denominator: 3) Divide, then reduce: 1 1 =

1 1 1 P Complex Rational Expressions LCD Method: Simplify by multiplying an LCD Simplify: 1) Find LCD for all denominators: 4= 2 2 2) Multiply all terms by LCD: 3) Simplify, then reduce as needed. 3= 3 LCD 2 2 3 12 31 24

+ 4 3 1 1 3 4 ( ) ( ) ( ) 1 1( ) 2 1 3 4 3 8 11 + 8 3 5

P Complex Rational Expressions LCD Method: Simplify by multiplying an LCD Simplify: 1) Find LCD for all denominators: 2) Multiply all terms by LCD: LCD 1 1 ( ) 1 1 1 (

) 2 () 1 ( 2 ) + 3) Simplify, then reduce as needed. 2 1 + 2 1 P Complex Rational Expressions LCD Method: Simplify by multiplying an LCD

Simplify: 1) Find LCD for all denominators: 2) Multiply all terms by LCD: 3) Simplify, then reduce as needed. LCD ( 1 1 ) 1 1( ) 1 11 ( )

1 1 P Practice and Complete HW 7.5 as scheduled. MATH 0322 Intermediate Algebra Unit 2 Rational Equations Section: 7.6 Rational Equations Rational equations are equations containing one or

more rational expressions. Examples: , Skills will you need to solve this type of equation: 1) 2) 3) 4) 5) 6) Factoring (Ch6) Determine restricted values (Ch7.1) Form an LCD (Ch7.4) Add or subtract rational expressions (Ch7.4) Use an LCD as a multiplier (Ch7.5) Solve an equation for a specified variable Rational Equations Steps to solve rational equations: 1) Factor denominators to find restricted values and LCD.

2) Multiply all terms by LCD and simplify. 3) Solve and check proposed solutions. (Remember, restricted values are always rejected as solutions to the equation.) Rational Equations Practice: Determine the restricted values only. A rational equation has restricted values if its denominators contain a variable. Restricted values cause division by zero. Does this rational equation have restricted values? Yes or No , therefore can be any real number. P Rational Equations Practice: Determine the restricted values only. Denominators contain a variable.

This rational equation has restricted value(s). What is the restricted value(s)? 0P Rational Equations Practice: Determine the restricted values only. Solve Solve Solve Restricted values: 4 4 ( + 4)( 4) 0 , , 40 4

4, 4P Rational Equations Practice: Solve the rational equation. 1) Restricted values and LCD 0 LCD 6 1 ()+() = 5() 1 2) Multiply LCD 2 3) Solve and Check

, + +6= 5 + 2 +5 +6=0 () +2( )=0 +3 +3=0 =2P =3P Rational Equations Practice: Solve the rational equation. 1) Restricted values and LCD

2 25 0 ( )() +5 5 0 +5 0 5 0 5 5 +5 0 5P LCD 5 0 5P (+5)( 5)P

Rational Equations Practice: Solve the rational equation. 2) Multiply LCD 11 4 3 + = ( + 5)( 5) +5 5 1 11 4 3 1 1 (+)( ) (+ )( (+)( ) + ) = 5

( + 5 )( 5 ) +5 1 1 1 11+ 4 ( 5)=3( +5) P Rational Equations Practice: Solve the rational equation. 3) Solve and Check 11+ 4 ( 5)=3( +5) 11+ 4 20=3 +15 4 9=3 +15 9=15 =24

5,5 Solution is . P Rational Equations Practice: A company that manufactures wheelchairs has fixed costs of $500,000. The average cost per wheelchair is C dollars for the company to produce x wheelchairs per month and is described by the given formula: How many wheelchairs per month can be produced at an average cost of $450 per wheelchair? Rational Equations In this problem, follow the steps used to solve a rational equation by first substituting for the

variable . 1) Restricted value(s) and LCD LCD Rational Equations 2) Multiply LCD 1 () () 1 3) Solve and Check Solution is wheelchairs.

P