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1 MATH 90 CHAPTER 6 Name:. 6.1 GCF and Factoring by Groups Need To Know Definitions How to factor by GCF How to factor by groups The Greatest Common Factor Factoring means to write a number as product. a polynomial means to a polynomial as a. The Greatest Common Factor of a polynomial is the largest monomial that divides each term of the polynomial. Examples Identify the GCF then FACTOR: 6x y y 3 4y 2 The Greatest Common Factor Factor each polynomial ab + ac You Try 25x x 3 7x x x 4-20x 8 12x 7 + 4x 6-35a 6 z a 4 z 7 21a 3 z 7 Scott Eckert pg. 1

2 Factoring by Groups Factor by Groups 1. No GCF and 4 terms Factor by Groups: 1) a 3 7a 2 + 4a 28 2) 8x x 2 14x 21 3) 20g 3 4g 35g end 6.2 Factoring Trinomials Need To Know Diamond puzzle Idea of reverse FOIL Factoring Trinomials Scott Eckert pg. 2

3 Diamond Puzzle A puzzle that builds mental skills needed for factoring. Find two numbers that: multiply to the top answer and add to the bottom answer. 6?? 5 20?? 9-9?? 0-18?? -7-36?? -5 The Foil Method in Reverse Multiply: (z 6)(z + 2) z 2 4z 12 Short Cut: FOIL F first terms mult. O outer terms mult. I inner terms mult. L last terms mult. Reverse FOIL 1. Write out parentheses ( )( ) 2. pair that multiplies to the last and adds to the middle term. 3. and check for a match on the add term. 4. Guess, Check O & I, Revise Factoring Trinomials x 2 + 5x + 6 z 2 7z + 6 Scott Eckert pg. 3

4 Factoring Trinomials w/ GCF 2a 2 2a 24 3z z z Factoring Trinomials w/ 2 Vars x 2 + 7xy + 12y 2 z 2 3za 10a 2 end 6.3 More Factoring Trinomials Need To Know Review diamond puzzle Methods of factoring Trinomials 1. Guess, check and revise 2. The Grouping Method (see the book for this technique) Scott Eckert pg. 4

5 Diamond Puzzle Limitations x 2 3x 10 2a 2 + 7a + 6 Factoring Trinomials w/ a 1 2a 2 + 7a + 6 Reverse FOIL 0. Standard Form; Factor GCF 1. Write out parentheses ( )( ) 2. Pick the sign pair that multiplies to the last and adds to the middle term Guess, Revise Factoring Trinomials w/ a 1 21y 2 70y 56 Reverse FOIL 0. Standard Form; Factor GCF 1. Write out parentheses ( )( ) 2. Pick the sign pair that multiplies to the last and adds to the middle term. 3. List all factorings of first term 4. List all factorings of last term 5. Guess, Check O & I, Revise Scott Eckert pg. 5

6 Conclusion Ways to Factor Polynomial 1. By Greatest Common Factor (GCF) 2. By Grouping 3. Factor Trinomials Guess, check and revise The Group Method (see book) end Scott Eckert pg. 6

7 6.4 Special Factoring Need To Know Recall some formulas Factoring the difference of two squares Factoring perfect square trinomials Formulas to Remember: 1. (A + B)(A B) = 2. (A + B) 2 = 3. (A B) 2 = Factoring a Difference of Squares Formulas to Know A 2 B 2 = (A + B)(A B) 9x 2 25 A 2 + 2AB + B 2 = (A + B) 2 A 2 2AB + B 2 =( A B) 2 49z 2 1 Factoring a Difference of Squares Formulas to Know A 2 B 2 = (A + B)(A B) A 2 + B 2 x A 2 + 2AB + B 2 = (A + B) 2 A 2 2AB + B 2 = ( A B) 2 a 4 16 Scott Eckert pg. 7

8 Factor- Perfect Square Trinomials Formulas to Know A 2 B 2 = (A + B)(A B) A 2 + B 2 CAN T Factor m m + 36 A 2 + 2AB + B 2 = (A + B) 2 A 2 2AB + B 2 = ( A B) 2 4x x + 9 Factor- Perfect Square Trinomials Formulas to Know A 2 B 2 = (A + B)(A B) 16x x A 2 + B 2 CAN T Factor A 2 + 2AB + B 2 = (A + B) 2 A 2 2AB + B 2 = ( A B) 2 18x 2 60x + 50 Conclusion Ways to Factor Polynomial 1. By Greatest Common Factor (GCF) 2. By Grouping 3. Factor Trinomials Guess, check and revise 4. Factoring with Formulas A 2 B 2 = (A + B)(A B) A 2 + B 2 CAN T Factor A 2 + 2AB + B 2 = (A + B) 2 A 2 2AB + B 2 = (A B) 2 Scott Eckert pg. 8

9 6.5 Special Factoring - CUBES Need To Know Factoring a Difference of Cubes Factoring a Sum of Cubes Using Formulas Factoring Cubes Factor x 3-64 Formulas for Factoring Cubes Factoring a Sum or Difference of Two Cubes When factoring a sum or difference of cubes, it can be helpful to remember that 2 3 = 8, 3 3 = 27, 4 3 = 64, 5 3 = 125, 6 3 = 216, 10 3 = 1000 The list of number: are called perfect cubes. Scott Eckert pg. 9

10 Factoring by Using Formulas Sum of cubes: A 3 + B 3 = Difference of cubes: A 3 B 3 = Difference of squares: A 2 B 2 = Sum of two squares NO FORMULA A 2 + B 2 can t be factored Scott Eckert pg. 10

11 6.6 Factoring Review Need To Know Factoring Review Practice Ways to Factor Based on Terms A. Check for GCF factoring (Always do first!) B. Look at the Number of Terms Two Terms Formulas A 2 B 2 = (A + B)(A B) A 2 + B 2 can NOT factor Three Terms Guess, check, and revise Formulas: A 2 + 2AB + B 2 = (A + B) 2 A 2-2AB + B 2 = (A - B) 2 Four Terms By Grouping C. Always Factor Completely Try to factor more. Scott Eckert pg. 11

12 Factoring Practice WAYS TO FACTOR A. Factor GCF B. Look at the Number of Terms Two Terms Use Formula A 2 B 2 = (A + B)(A B) A 2 + B 2 can t factor A 3 B 3 = (A B)(A 2 + AB + B 2 ) Three Terms 1. Guess, check, and revise 2. Formulas A 2 + 2AB + B 2 = (A + B) 2 A 2 2AB + B 2 = (A B) 2 Four Terms By Grouping Method C. Always Factor Completely Try to factor more! w a 2 b 2 ab 2 2x x x 3 Factoring Practice WAYS TO FACTOR A. Factor GCF B. Look at the Number of Terms Two Terms Use Formula A 2 B 2 = (A + B)(A B) A 2 + B 2 can t factor A 3 B 3 = (A B)(A 2 + AB + B 2 ) Three Terms 1. Guess, check, and revise 2. Formulas A 2 + 2AB + B 2 = (A + B) 2 A 2 2AB + B 2 = (A B) 2 Four Terms By Grouping Method C. Always Factor Completely Try to factor more! -4x x x 2 2a 5 b + 6a 4 b + 2a 3 b 3ab + 9a + 2b + 6 Scott Eckert pg. 12

13 6.7 Solving Quadratic Equation Need To Know Vocabulary and facts Solving quadratic equations Vocabulary Definition The degree of a polynomial in one variable is the exponent from the term with the highest power. Fact The degree of an equation. The Principle of Zero Products For real numbers a and b, Solve Polynomial Equations Solve for x: (x + 3)(x + 10) = 0 Solve for z: 6(5z 3)(z + 8) = 0 Scott Eckert pg. 13

14 Vocabulary Definition a quadratic equation is one that can be put in the form ax 2 + bx + c = 0 where a, b, & c are real numb. (a 0). ax 2 + bx + c = 0 is a quadratic in. Linear Term Quadratic Term Constant Term Solve Quadratic Equations Steps to Solve Q.E Check solutions Solve for x: x 2 + 6x + 8 = 0 Solve Quadratic Equations Steps to Solve Q.E. 1. Set up must = 0 2. Factor completely 3. Solve each factor for zero 4. Check solutions Solve for x: 2x 2 + 5x = 3 Scott Eckert pg. 14

15 Solve Quadratic Equations Solve for w: w 2 (2w 1) = 3w Solve for x: (2x-5)(3x 2 +29x+56)=0 end Scott Eckert pg. 15

16 6.8 Applications Need To Know Recall guide lines to solve word problems Recall tool to solve word problems Solve word problems with 2 unknowns Guide Lines for Word Problems Blueprint for Solving 1. Read and understand the problem (# of unknowns) 2. Assign variables and write down the meaning of the variable 3. Write an equation 4. Solve the equation 5. Write down your answer using a complete sentence 6. Reread and check your solution Tools to Reveal the Equation 1. Use keywords 2. Draw a picture 3. Make up a simpler problem 4. Make tables of numbers and look for patterns 5. Use charts to organize your information 6. Make a guess 7. Use a verbal model Apply A number is two less than its square. Find all such numbers. Steps 1. Familiarize 2. Translate 3. Carry out 4. Check 5. State answer Tools 1. Keywords 2. Drawing 3. Simpler problem 4. Tables/Patterns 5. Charts 6. Guess 7. Verbal Model Scott Eckert pg. 16

17 Apply The product of two consecutive odd integers is 63. Find the integers. Steps 1. Familiarize 2. Translate 3. Carry out 4. Check 5. State answer Tools 1. Keywords 2. Drawing 3. Simpler problem 4. Tables/Patterns 5. Charts 6. Guess 7. Verbal Model Apply The length of a rectangle is 2 more than twice the width. The area is 60 in 2. Find the dimensions. Steps 1. Familiarize 2. Translate 3. Carry out 4. Check 5. State answer Tools 1. Keywords 2. Drawing 3. Simpler problem 4. Tables/Patterns 5. Charts 6. Guess 7. Verbal Model Scott Eckert pg. 17

18 Pythagoras and The Theorem Pythagorean Theorem: In any right triangle, if a and b are the lengths of the legs and c is the length of the hypotenuse, a c. b Apply Deb s garden was a right triangle. One leg of the triangle was formed by a 400-ft wall. The hypotenuse of the triangle was 200 ft longer than the other leg. What are the dimensions? Steps 1. Familiarize 2. Translate 3. Carry out 4. Check 5. State answer Tools 1. Keywords 2. Drawing 3. Simpler problem 4. Tables/Patterns 5. Charts 6. Guess 7. Verbal Model Scott Eckert pg. 18

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