Factor and Solve Polynomial Equations. In Chapter 4, you learned how to factor the following types of quadratic expressions.


 Toby Allison
 3 years ago
 Views:
Transcription
1 5.4 Factor and Solve Polynomial Equations Before You factored and solved quadratic equations. Now You will factor and solve other polynomial equations. Why? So you can find dimensions of archaeological ruins, as in Ex. 58. Key Vocabulary factored completely factor by grouping quadratic form In Chapter 4, you learned how to factor the following types of quadratic expressions. Type Example General trinomial 2x 2 2 3x (2x 1 5)(x 2 4) Perfect square trinomial x 2 1 8x (x 1 4) 2 Difference of two squares 9x (3x 1 1)(3x 2 1) Common monomial factor 8x x 5 4x(2x 1 5) You can also factor polynomials with degree greater than 2. Some of these polynomials can be factored completely using techniques learned in Chapter 4. KEY CONCEPT For Your Notebook Factoring Polynomials Definition A factorable polynomial with integer coefficients is factored completely if it is written as a product of unfactorable polynomials with integer coefficients. Examples 2(x 1 1)(x 2 4) and 5x 2 (x 2 2 3) are factored completely. 3x(x 2 2 4) is not factored completely because x can be factored as (x 1 2)(x 2 2). E XAMPLE 1 Find a common monomial factor Factor the polynomial completely. a. x 3 1 2x x 5 x(x 2 1 2x 2 15) Factor common monomial. 5 x(x 1 5)(x 2 3) Factor trinomial. b. 2y y 3 5 2y 3 (y 2 2 9) Factor common monomial. 5 2y 3 (y 1 3)(y 2 3) Difference of two squares c. 4z z z 2 5 4z 2 (z 2 2 4z 1 4) Factor common monomial. 5 4z 2 (z 2 2) 2 Perfect square trinomial 5.4 Factor and Solve Polynomial Equations 353
2 FACTORING PATTERNS In part (b) of Example 1, the special factoring pattern for the difference of two squares is used to factor the expression completely. There are also factoring patterns that you can use to factor the sum or difference of two cubes. KEY CONCEPT For Your Notebook Special Factoring Patterns Sum of Two Cubes Example a 3 1 b 3 5 (a 1 b)(a 2 2 ab 1 b 2 ) 8x (2x) (2x 1 3)(4x 2 2 6x 1 9) Difference of Two Cubes Example a 3 2 b 3 5 (a 2 b)(a 2 1 ab 1 b 2 ) 64x (4x) (4x 2 1)(16x 2 1 4x 1 1) E XAMPLE 2 Factor the sum or difference of two cubes Factor the polynomial completely. a. x x Sum of two cubes 5 (x 1 4)(x 2 2 4x 1 16) b. 16z z 2 5 2z 2 (8z ) Factor common monomial. 5 2z 2 F (2z) G Difference of two cubes 5 2z 2 (2z 2 5)(4z z 1 25) GUIDED PRACTICE for Examples 1 and 2 Factor the polynomial completely. 1. x 3 2 7x x 2. 3y y b b 2 4. w FACTORING BY GROUPING For some polynomials, you can factor by grouping pairs of terms that have a common monomial factor. The pattern for factoring by grouping is shown below. ra 1 rb 1 sa 1 sb 5 r(a 1 b) 1 s(a 1 b) 5 (r 1 s)(a 1 b) E XAMPLE 3 Factor by grouping AVOID ERRORS An expression is not factored completely until all factors, such as x , cannot be factored further. Factor the polynomial x 3 2 3x x 1 48 completely. x 3 2 3x x x 2 (x 2 3) 2 16(x 2 3) Factor by grouping. 5 (x )(x 2 3) Distributive property 5 (x 1 4)(x 2 4)(x 2 3) Difference of two squares 354 Chapter 5 Polynomials and Polynomial Functions
3 QUADRATIC FORM An expression of the form au 2 1 bu 1 c, where u is any expression in x, is said to be in quadratic form. The factoring techniques you studied in Chapter 4 can sometimes be used to factor such expressions. E XAMPLE 4 Factor polynomials in quadratic form IDENTIFY QUADRATIC FORM The expression 16x is in quadratic form because it can be written as u where u 5 4x 2. Factor completely: (a) 16x and (b) 2p p p 2. a. 16x (4x 2 ) Write as difference of two squares. 5 (4x 2 1 9)(4x 2 2 9) Difference of two squares 5 (4x 2 1 9)(2x 1 3)(2x 2 3) Difference of two squares b. 2p p p 2 5 2p 2 (p 6 1 5p 3 1 6) Factor common monomial. 5 2p 2 (p 3 1 3)(p 3 1 2) Factor trinomial in quadratic form. GUIDED PRACTICE for Examples 3 and 4 Factor the polynomial completely. 5. x 3 1 7x 2 2 9x g t t t 2 SOLVING POLYNOMIAL EQUATIONS In Chapter 4, you learned how to use the zero product property to solve factorable quadratic equations. You can extend this technique to solve some higherdegree polynomial equations. E XAMPLE 5 Standardized Test Practice What are the realnumber solutions of the equation 3x x 5 18x 3? A 0, 1, 3, 5 B 21, 0, 1 C 0, 1, Ï 5 D 2 Ï 5, 21, 0, 1, Ï 5 Solution 3x x 5 18x 3 Write original equation. AVOID ERRORS Do not divide each side of an equation by a variable or a variable expression, such as 3x. Doing so will result in the loss of solutions. 3x x x 5 0 Write in standard form. 3x(x 4 2 6x 2 1 5) 5 0 Factor common monomial. 3x(x 2 2 1)(x 2 2 5) 5 0 Factor trinomial. 3 x(x 1 1)(x 2 1)(x 2 2 5) 5 0 Difference of two squares x 5 0, x 5 21, x 5 1, x 5 Ï 5, or x 5 2 Ï 5 Zero product property c The correct answer is D. A B C D GUIDED PRACTICE for Example 5 Find the realnumber solutions of the equation. 8. 4x x x x x 5 14x x x x Factor and Solve Polynomial Equations 355
4 E XAMPLE 6 Solve a polynomial equation CITY PARK You are designing a marble basin that will hold a fountain for a city park. The basin s sides and bottom should be 1 foot thick. Its outer length should be twice its outer width and outer height. What should the outer dimensions of the basin be if it is to hold 36 cubic feet of water? ANOTHER WAY For alternative methods to solving the problem in Example 6, turn to page 360 for the Problem Solving Workshop. Solution Volume (cubic feet) 5 Interior length (feet) p Interior width (feet) p Interior height (feet) 36 5 (2x 2 2) p (x 2 2) p (x 2 1) 36 5 (2x 2 2)(x 2 2)(x 2 1) Write equation x 3 2 8x x 2 40 Write in standard form x 2 (x 2 4) 1 10(x 2 4) Factor by grouping. 0 5 (2x )(x 2 4) Distributive property c The only real solution is x 5 4. The basin is 8 ft long, 4 ft wide, and 4 ft high. GUIDED PRACTICE for Example WHAT IF? In Example 6, what should the basin s dimensions be if it is to hold 128 cubic feet of water and have outer length 6x, width 3x, and height x? 5.4 EXERCISES SKILL PRACTICE HOMEWORK KEY 5 WORKEDOUT SOLUTIONS on p. WS1 for Exs. 7, 23, and 61 5 STANDARDIZED TEST PRACTICE Exs. 2, 9, 41, 63, and VOCABULARY The expression 8x x is in? form because it can be written as 2u 2 1 5u 2 3 where u 5 2x WRITING What condition must the factorization of a polynomial satisfy in order for the polynomial to be factored completely? EXAMPLE 1 on p. 353 for Exs. 3 9 MONOMIAL FACTORS Factor the polynomial completely x x 4. 30b b 2 5. c 3 1 9c c 6. z 3 2 6z z 7. 3y y m m 4 1 9m 3 9. MULTIPLE CHOICE What is the complete factorization of 2x x 3? A 2x 3 (x 1 2)(x 2 2)(x 2 1 4) B 2x 3 (x 2 1 2)(x 2 2 2) C 2x 3 (x 2 1 4) 2 D 2x 3 (x 1 2) 2 (x 2 2) Chapter 5 Polynomials and Polynomial Functions
5 EXAMPLE 2 on p. 354 for Exs EXAMPLE 3 on p. 354 for Exs EXAMPLE 4 on p. 355 for Exs EXAMPLE 5 on p. 355 for Exs SUM OR DIFFERENCE OF CUBES Factor the polynomial completely. 10. x y m n a c w z FACTORING BY GROUPING Factor the polynomial completely. 18. x 3 1 x 2 1 x y 3 2 7y 2 1 4y n 3 1 5n 2 2 9n m 3 2 m 2 1 9m s s 2 2 s c 3 1 8c 2 2 9c 2 18 QUADRATIC FORM Factor the polynomial completely. 24. x a 4 1 7a s 4 2 s z 5 2 2z m m 4 1 m x x x ERROR ANALYSIS Describe and correct the error in finding all realnumber solutions x (2x 1 3)(4x 2 1 6x 1 9) 5 0 x x x 5 0 3x(x ) 5 0 x x 5 24 or x 5 4 SOLVING EQUATIONS Find the realnumber solutions of the equation. 32. y 3 2 5y s s 34. g 3 1 3g 2 2 g m 3 1 6m 2 2 4m w w z z b b b x 6 2 4x 4 2 9x p p MULTIPLE CHOICE What are the realnumber solutions of the equation 3x x 2 1 9x 5 x 3? A 21, 0, 3 B 23, 0, 3 C 23, 0, 1 3, 3 D 23, 2 1 3, 0, 3 CHOOSING A METHOD Factor the polynomial completely using any method x x x 43. n 4 2 4n b b a a a c c c d d x x y y y z 5 2 3z z 1 48 GEOMETRY Find the possible value(s) of x. 51. Area Volume Volume 5 125π x 2 4 2x 2 5 x 1 4 2x 3x 3x 1 2 x 2 1 CHOOSING A METHOD Factor the polynomial completely using any method. 54. x 3 y ac 2 1 bc 2 2 7ad 2 2 bd x 2n 2 2x n CHALLENGE Factor a 5 b 2 2 a 2 b 4 1 2a 4 b 2 2ab 3 1 a 3 2 b 2 completely. 5.4 Factor and Solve Polynomial Equations 357
6 PROBLEM SOLVING EXAMPLE 6 on p. 356 for Exs ARCHAEOLOGY At the ruins of Caesarea, archaeologists discovered a huge hydraulic concrete block with a volume of 945 cubic meters. The block s dimensions are x meters high by 12x 2 15 meters long by 12x 2 21 meters wide. What is the height of the block? LEBANON Caesarea SYRIA EGYPT ISRAEL JORDAN 59. CHOCOLATE MOLD You are designing a chocolate mold shaped like a hollow rectangular prism for a candy manufacturer. The mold must have a thickness of 1 centimeter in all dimensions. The mold s outer dimensions should also be in the ratio 1: 3: 6. What should the outer dimensions of the mold be if it is to hold 112 cubic centimeters of chocolate? 60. MULTISTEP PROBLEM A production crew is assembling a threelevel platform inside a stadium for a performance. The platform has the dimensions shown in the diagrams, and has a total volume of 1250 cubic feet. 4x 6x 8x 2x 4x 6x x x x a. Write Expressions What is the volume, in terms of x, of each of the three levels of the platform? b. Write an Equation Use what you know about the total volume to write an equation involving x. c. Solve Solve the equation from part (b). Use your solution to calculate the dimensions of each of the three levels of the platform. 61. SCULPTURE Suppose you have 250 cubic inches of clay with which to make a sculpture shaped as a rectangular prism. You want the height and width each to be 5 inches less than the length. What should the dimensions of the prism be? 62. MANUFACTURING A manufacturer wants to build a rectangular stainless steel tank with a holding capacity of 670 gallons, or about cubic feet. The tank s walls will be one half inch thick, and about 6.42 cubic feet of steel will be used for the tank. The manufacturer wants the outer dimensions of the tank to be related as follows: The width should be 2 feet less than the length. The height should be 8 feet more than the length. What should the outer dimensions of the tank be? x x 1 8 x WORKEDOUT SOLUTIONS 358 Chapter 5 Polynomials on p. WS1 and Polynomial Functions 5 STANDARDIZED TEST PRACTICE
7 63. SHORT RESPONSE A platform shaped like a rectangular prism has dimensions x 2 2 feet by 3 2 2x feet by 3x 1 4 feet. Explain why the volume of the platform cannot be 7 cubic feet EXTENDED RESPONSE In 2000 B.C., the Babylonians solved polynomial equations using tables of values. One such table gave values of y 3 1 y 2. To be able to use this table, the Babylonians sometimes had to manipulate the equation, as shown below. ax 3 1 bx 2 5 c a 3 x 3 1 a2 x 2 b 3 b 2 1 ax b ax 5 a2 c b 3 b a2 c b 3 Original equation Multiply each side by a2 b 3. Rewrite cubes and squares. They then found a2 c b in the 3 y3 1 y 2 column of the table. Because the corresponding yvalue was y 5 ax by, they could conclude that x 5 b a. a. Calculate y 3 1 y 2 for y 5 1, 2, 3,..., 10. Record the values in a table. b. Use your table and the method described above to solve x 3 1 2x c. Use your table and the method described above to solve 3x 3 1 2x d. How can you modify the method described above for equations of the form ax 4 1 bx 3 5 c? 65. CHALLENGE Use the diagram to complete parts (a) (c). a. Explain why a 3 2 b 3 is equal to the sum of the volumes of solid I, solid II, and solid III. b. Write an algebraic expression for the volume of each of the three solids. Leave your expressions in factored form. c. Use the results from parts (a) and (b) to derive the factoring pattern for a 3 2 b 3 given on page 354. II b I a b III b a a MIXED REVIEW Graph the function. 66. f(x) 5 22 x (p. 123) 67. y x2 1 4x 1 5 (p. 236) 68. y 5 3(x 1 4) (p. 245) 69. f(x) 5 x 3 2 2x 2 5 (p. 337) Graph the inequality in a coordinate plane. (p. 132) 70. y 2x y > 25 2 x 72. y < 0.5x x 1 12y x 2 9y x y > 5 PREVIEW Prepare for Lesson 5.5 in Exs Use synthetic substitution to evaluate the polynomial function for the given value of x. (p. 337) 76. f(x) 5 5x 4 2 3x 3 1 4x 2 2 x 1 10; x f(x) 5 23x 5 1 x 3 2 6x 2 1 2x 1 4; x f(x) 5 5x 5 2 4x x ; x f(x) 5 26x 4 1 9x 2 15; x 5 4 EXTRA PRACTICE for Lesson 5.4, p ONLINE Factor and QUIZ Solve Polynomial at classzone.com Equations 359
8 LESSON 5.4 Using ALTERNATIVE METHODS Another Way to Solve Example 6, page 356 MULTIPLE REPRESENTATIONS In Example 6 on page 356, you solved a polynomial equation by factoring. You can also solve a polynomial equation using a table or a graph. P ROBLEM CITY PARK You are designing a marble basin that will hold a fountain for a city park. The basin s sides and bottom should be 1 foot thick. Its outer length should be twice its outer width and outer height. What should the outer dimensions of the basin be if it is to hold 36 cubic feet of water? M ETHOD 1 Using a Table One alternative approach is to write a function for the volume of the basin and make a table of values for the function. Using the table, you can find the value of x that makes the volume of the basin 36 cubic feet. STEP 1 Write the function. From the diagram, you can see that the volume y of water the basin can hold is given by this function: y 5 (2x 2 2)(x 2 2)(x 2 1) STEP 2 Make a table of values for the function. Use only positive values of x because the basin s dimensions must be positive. STEP 3 Identify the value of x for which y The table shows that y 5 36 when x 5 4. X Y1= Y1 X Y1= Y1 c The volume of the basin is 36 cubic feet when x is 4 feet. So, the outer dimensions of the basin should be as follows: Length 5 2x 5 8 feet Width 5 x 5 4 feet Height 5 x 5 4 feet 360 Chapter 5 Polynomials and Polynomial Functions
9 M ETHOD 2 Using a Graph Another approach is to make a graph. You can use the graph to find the value of x that makes the volume of the basin 36 cubic feet. STEP 1 Write the function. From the diagram, you can see that the volume y of water the basin can hold is given by this function: y 5 (2x 2 2)(x 2 2)(x 2 1) STEP 2 Graph the equations y 5 36 and y 5 (x 2 1)(2x 2 2)(x 2 2). Choose a viewing window that shows the intersection of the graphs. STEP 3 Identify the coordinates of the intersection point. On a graphing calculator, you can use the intersect feature. The intersection point is (4, 36). Intersection X=4 Y=36 c The volume of the basin is 36 cubic feet when x is 4 feet. So, the outer dimensions of the basin should be as follows: Length 5 2x 5 8 feet Width 5 x 5 4 feet Height 5 x 5 4 feet P RACTICE SOLVING EQUATIONS Solve the polynomial equation using a table or using a graph. 1. x 3 1 4x 2 2 8x x 3 2 9x x x x 2 1 3x x 4 1 x x 2 2 8x x 4 1 2x 3 1 6x x x 4 1 4x 3 1 8x 2 1 4x x x x x WHAT IF? In the problem on page 360, suppose the basin is to hold 200 cubic feet of water. Find the outer dimensions of the basin using a table and using a graph. 9. PACKAGING A factory needs a box that has a volume of 1728 cubic inches. The width should be 4 inches less than the height, and the length should be 6 inches greater than the height. Find the dimensions of the box using a table and using a graph. 10. AGRICULTURE From 1970 to 2002, the average yearly pineapple consumption P (in pounds) per person in the United States can be modeled by the function P(x) x x x x where x is the number of years since In what year was the pineapple consumption about 9.97 pounds per person? Solve the problem using a table and a graph. Using Alternative Methods 361
Factor Polynomials Completely
9.8 Factor Polynomials Completely Before You factored polynomials. Now You will factor polynomials completely. Why? So you can model the height of a projectile, as in Ex. 71. Key Vocabulary factor by grouping
More informationFactoring. Factoring Polynomial Equations. Special Factoring Patterns. Factoring. Special Factoring Patterns. Special Factoring Patterns
Factoring Factoring Polynomial Equations Ms. Laster Earlier, you learned to factor several types of quadratic expressions: General trinomial  2x 25x12 = (2x + 3)(x  4) Perfect Square Trinomial  x
More information6.4 Factoring Polynomials
Name Class Date 6.4 Factoring Polynomials Essential Question: What are some ways to factor a polynomial, and how is factoring useful? Resource Locker Explore Analyzing a Visual Model for Polynomial Factorization
More informationMonomial. 5 1 x A sum is not a monomial. 2 A monomial cannot have a. x 21. degree. 2x 3 1 x 2 2 5x Rewrite a polynomial
9.1 Add and Subtract Polynomials Before You added and subtracted integers. Now You will add and subtract polynomials. Why? So you can model trends in recreation, as in Ex. 37. Key Vocabulary monomial degree
More information6.3 FACTORING ax 2 bx c WITH a 1
290 (6 14) Chapter 6 Factoring e) What is the approximate maximum revenue? f) Use the accompanying graph to estimate the price at which the revenue is zero. y Revenue (thousands of dollars) 300 200 100
More informationSolve Quadratic Equations by the Quadratic Formula. The solutions of the quadratic equation ax 2 1 bx 1 c 5 0 are. Standardized Test Practice
10.6 Solve Quadratic Equations by the Quadratic Formula Before You solved quadratic equations by completing the square. Now You will solve quadratic equations using the quadratic formula. Why? So you can
More informationUse Square Roots to Solve Quadratic Equations
10.4 Use Square Roots to Solve Quadratic Equations Before You solved a quadratic equation by graphing. Now You will solve a quadratic equation by finding square roots. Why? So you can solve a problem about
More information7.2 Quadratic Equations
476 CHAPTER 7 Graphs, Equations, and Inequalities 7. Quadratic Equations Now Work the Are You Prepared? problems on page 48. OBJECTIVES 1 Solve Quadratic Equations by Factoring (p. 476) Solve Quadratic
More informationDefinitions 1. A factor of integer is an integer that will divide the given integer evenly (with no remainder).
Math 50, Chapter 8 (Page 1 of 20) 8.1 Common Factors Definitions 1. A factor of integer is an integer that will divide the given integer evenly (with no remainder). Find all the factors of a. 44 b. 32
More informationFactoring Polynomials
UNIT 11 Factoring Polynomials You can use polynomials to describe framing for art. 396 Unit 11 factoring polynomials A polynomial is an expression that has variables that represent numbers. A number can
More informationMATH 90 CHAPTER 6 Name:.
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
More informationexpression is written horizontally. The Last terms ((2)( 4)) because they are the last terms of the two polynomials. This is called the FOIL method.
A polynomial of degree n (in one variable, with real coefficients) is an expression of the form: a n x n + a n 1 x n 1 + a n 2 x n 2 + + a 2 x 2 + a 1 x + a 0 where a n, a n 1, a n 2, a 2, a 1, a 0 are
More informationSPECIAL PRODUCTS AND FACTORS
CHAPTER 442 11 CHAPTER TABLE OF CONTENTS 111 Factors and Factoring 112 Common Monomial Factors 113 The Square of a Monomial 114 Multiplying the Sum and the Difference of Two Terms 115 Factoring the
More information6.1 Add & Subtract Polynomial Expression & Functions
6.1 Add & Subtract Polynomial Expression & Functions Objectives 1. Know the meaning of the words term, monomial, binomial, trinomial, polynomial, degree, coefficient, like terms, polynomial funciton, quardrtic
More informationNSM100 Introduction to Algebra Chapter 5 Notes Factoring
Section 5.1 Greatest Common Factor (GCF) and Factoring by Grouping Greatest Common Factor for a polynomial is the largest monomial that divides (is a factor of) each term of the polynomial. GCF is the
More informationCopy in your notebook: Add an example of each term with the symbols used in algebra 2 if there are any.
Algebra 2  Chapter Prerequisites Vocabulary Copy in your notebook: Add an example of each term with the symbols used in algebra 2 if there are any. P1 p. 1 1. counting(natural) numbers  {1,2,3,4,...}
More informationFactoring Quadratic Expressions
Factoring the trinomial ax 2 + bx + c when a = 1 A trinomial in the form x 2 + bx + c can be factored to equal (x + m)(x + n) when the product of m x n equals c and the sum of m + n equals b. (Note: the
More information1.3 Polynomials and Factoring
1.3 Polynomials and Factoring Polynomials Constant: a number, such as 5 or 27 Variable: a letter or symbol that represents a value. Term: a constant, variable, or the product or a constant and variable.
More informationComplete this equation: 2m = 2? PROPERTIES OF EXPONENTS PRODUCT OF POWERS PROPERTY POWER OF A POWER PROPERTY POWER OF A PRODUCT PROPERTY
Page of 6 6. Using Properties of Eponents What you should learn GOAL Use properties of eponents to evaluate and simplify epressions involving powers. GOAL Use eponents and scientific notation to solve
More informationACTIVITY: Multiplying Binomials Using Algebra Tiles. Work with a partner. Six different algebra tiles are shown below.
7.3 Multiplying Polynomials How can you multiply two binomials? 1 ACTIVITY: Multiplying Binomials Using Algebra Tiles Work with a partner. Six different algebra tiles are shown below. 1 1 x x x x Write
More informationName Intro to Algebra 2. Unit 1: Polynomials and Factoring
Name Intro to Algebra 2 Unit 1: Polynomials and Factoring Date Page Topic Homework 9/3 2 Polynomial Vocabulary No Homework 9/4 x In Class assignment None 9/5 3 Adding and Subtracting Polynomials Pg. 332
More informationSection 6.1 Factoring Expressions
Section 6.1 Factoring Expressions The first method we will discuss, in solving polynomial equations, is the method of FACTORING. Before we jump into this process, you need to have some concept of what
More informationMth 95 Module 2 Spring 2014
Mth 95 Module Spring 014 Section 5.3 Polynomials and Polynomial Functions Vocabulary of Polynomials A term is a number, a variable, or a product of numbers and variables raised to powers. Terms in an expression
More informationEvaluate and Simplify Algebraic Expressions
1.2 Evaluate and Simplify Algebraic Expressions Before You studied properties of real numbers. Now You will evaluate and simplify expressions involving real numbers. Why? So you can estimate calorie use,
More informationUNIT TWO POLYNOMIALS MATH 421A 22 HOURS. Revised May 2, 00
UNIT TWO POLYNOMIALS MATH 421A 22 HOURS Revised May 2, 00 38 UNIT 2: POLYNOMIALS Previous Knowledge: With the implementation of APEF Mathematics at the intermediate level, students should be able to: 
More informationx n = 1 x n In other words, taking a negative expoenent is the same is taking the reciprocal of the positive expoenent.
Rules of Exponents: If n > 0, m > 0 are positive integers and x, y are any real numbers, then: x m x n = x m+n x m x n = xm n, if m n (x m ) n = x mn (xy) n = x n y n ( x y ) n = xn y n 1 Can we make sense
More informationFACTORING POLYNOMIALS
296 (540) Chapter 5 Exponents and Polynomials where a 2 is the area of the square base, b 2 is the area of the square top, and H is the distance from the base to the top. Find the volume of a truncated
More informationFactoring Trinomials: The ac Method
6.7 Factoring Trinomials: The ac Method 6.7 OBJECTIVES 1. Use the ac test to determine whether a trinomial is factorable over the integers 2. Use the results of the ac test to factor a trinomial 3. For
More informationFactoring Polynomials
Factoring Polynomials Factoring Factoring is the process of writing a polynomial as the product of two or more polynomials. The factors of 6x 2 x 2 are 2x + 1 and 3x 2. In this section, we will be factoring
More informationPolynomial Degree and Finite Differences
CONDENSED LESSON 7.1 Polynomial Degree and Finite Differences In this lesson you will learn the terminology associated with polynomials use the finite differences method to determine the degree of a polynomial
More information10 7, 8. 2. 6x + 30x + 36 SOLUTION: 89 Perfect Squares. The first term is not a perfect square. So, 6x + 30x + 36 is not a perfect square trinomial.
Squares Determine whether each trinomial is a perfect square trinomial. Write yes or no. If so, factor it. 1.5x + 60x + 36 SOLUTION: The first term is a perfect square. 5x = (5x) The last term is a perfect
More informationFactoring Polynomials
Factoring a Polynomial Expression Factoring a polynomial is expressing the polynomial as a product of two or more factors. Simply stated, it is somewhat the reverse process of multiplying. To factor polynomials,
More information10.1. Solving Quadratic Equations. Investigation: Rocket Science CONDENSED
CONDENSED L E S S O N 10.1 Solving Quadratic Equations In this lesson you will look at quadratic functions that model projectile motion use tables and graphs to approimate solutions to quadratic equations
More informationAlgebra Tiles Activity 1: Adding Integers
Algebra Tiles Activity 1: Adding Integers NY Standards: 7/8.PS.6,7; 7/8.CN.1; 7/8.R.1; 7.N.13 We are going to use positive (yellow) and negative (red) tiles to discover the rules for adding and subtracting
More information1.3 Algebraic Expressions
1.3 Algebraic Expressions A polynomial is an expression of the form: a n x n + a n 1 x n 1 +... + a 2 x 2 + a 1 x + a 0 The numbers a 1, a 2,..., a n are called coefficients. Each of the separate parts,
More informationAlgebra Unit 6 Syllabus revised 2/27/13 Exponents and Polynomials
Algebra Unit 6 Syllabus revised /7/13 1 Objective: Multiply monomials. Simplify expressions involving powers of monomials. Preassessment: Exponents, Fractions, and Polynomial Expressions Lesson: Pages
More informationTopic: Special Products and Factors Subtopic: Rules on finding factors of polynomials
Quarter I: Special Products and Factors and Quadratic Equations Topic: Special Products and Factors Subtopic: Rules on finding factors of polynomials Time Frame: 20 days Time Frame: 3 days Content Standard:
More informationFactoring and Applications
Factoring and Applications What is a factor? The Greatest Common Factor (GCF) To factor a number means to write it as a product (multiplication). Therefore, in the problem 48 3, 4 and 8 are called the
More informationA.3. Polynomials and Factoring. Polynomials. What you should learn. Definition of a Polynomial in x. Why you should learn it
Appendi A.3 Polynomials and Factoring A23 A.3 Polynomials and Factoring What you should learn Write polynomials in standard form. Add,subtract,and multiply polynomials. Use special products to multiply
More informationStudy Guide and Review  Chapter 8
Study Guide Review  Chapter 8 Solve each equation. Check your solutions. 41. 6x 2 = 12x Factor the trinomial using the Zero Product Property. 43. 3x 2 = 5x Factor the trinomial using the Zero Product
More informationexpression undefined is called an excluded value. For example, } is x 2 3 undefined when x 5 3. So, 3 is an excluded value. Find excluded values
1.4 Simplify Rational Expressions Before You simplified polynomials. Now You will simplify rational expressions. Why So you can model a cost over time, as in Example. Key Vocabulary rational expression
More informationALGEBRA 2: 4.1 Graph Quadratic Functions in Standard Form
ALGEBRA 2: 4.1 Graph Quadratic Functions in Standard Form Goal Graph quadratic functions. VOCABULARY Quadratic function A function that can be written in the standard form y = ax 2 + bx+ c where a 0 Parabola
More informationVeterans Upward Bound Algebra I Concepts  Honors
Veterans Upward Bound Algebra I Concepts  Honors Brenda Meery Kaitlyn Spong Say Thanks to the Authors Click http://www.ck12.org/saythanks (No sign in required) www.ck12.org Chapter 6. Factoring CHAPTER
More informationSolving Quadratic Equations by Completing the Square
9. Solving Quadratic Equations by Completing the Square 9. OBJECTIVES 1. Solve a quadratic equation by the square root method. Solve a quadratic equation by completing the square. Solve a geometric application
More information1.6. Solve Linear Inequalities E XAMPLE 1 E XAMPLE 2. Graph simple inequalities. Graph compound inequalities
.6 Solve Linear Inequalities Before You solved linear equations. Now You will solve linear inequalities. Why? So you can describe temperature ranges, as in Ex. 54. Key Vocabulary linear inequality compound
More informationVOLUME of Rectangular Prisms Volume is the measure of occupied by a solid region.
Math 6 NOTES 7.5 Name VOLUME of Rectangular Prisms Volume is the measure of occupied by a solid region. **The formula for the volume of a rectangular prism is:** l = length w = width h = height Study Tip:
More informationFactoring, Solving. Equations, and Problem Solving REVISED PAGES
05W4801AM1.qxd 8/19/08 8:45 PM Page 241 Factoring, Solving Equations, and Problem Solving 5 5.1 Factoring by Using the Distributive Property 5.2 Factoring the Difference of Two Squares 5.3 Factoring
More informationPOLYNOMIAL FUNCTIONS
POLYNOMIAL FUNCTIONS Polynomial Division.. 314 The Rational Zero Test.....317 Descarte s Rule of Signs... 319 The Remainder Theorem.....31 Finding all Zeros of a Polynomial Function.......33 Writing a
More informationSect 6.7  Solving Equations Using the Zero Product Rule
Sect 6.7  Solving Equations Using the Zero Product Rule 116 Concept #1: Definition of a Quadratic Equation A quadratic equation is an equation that can be written in the form ax 2 + bx + c = 0 (referred
More informationMATH 65 NOTEBOOK CERTIFICATIONS
MATH 65 NOTEBOOK CERTIFICATIONS Review Material from Math 60 2.5 4.3 4.4a Chapter #8: Systems of Linear Equations 8.1 8.2 8.3 Chapter #5: Exponents and Polynomials 5.1 5.2a 5.2b 5.3 5.4 5.5 5.6a 5.7a 1
More informationFactoring a Difference of Two Squares. Factoring a Difference of Two Squares
284 (6 8) Chapter 6 Factoring 87. Tomato soup. The amount of metal S (in square inches) that it takes to make a can for tomato soup is a function of the radius r and height h: S 2 r 2 2 rh a) Rewrite this
More informationAlgebra I Vocabulary Cards
Algebra I Vocabulary Cards Table of Contents Expressions and Operations Natural Numbers Whole Numbers Integers Rational Numbers Irrational Numbers Real Numbers Absolute Value Order of Operations Expression
More informationSolving Quadratic Equations
9.3 Solving Quadratic Equations by Using the Quadratic Formula 9.3 OBJECTIVES 1. Solve a quadratic equation by using the quadratic formula 2. Determine the nature of the solutions of a quadratic equation
More informationPolynomial. Functions. 6A Operations with Polynomials. 6B Applying Polynomial. Functions. You can use polynomials to predict the shape of containers.
Polynomial Functions 6A Operations with Polynomials 61 Polynomials 6 Multiplying Polynomials 63 Dividing Polynomials Lab Explore the Sum and Difference of Two Cubes 64 Factoring Polynomials 6B Applying
More informationLesson 9.1 Solving Quadratic Equations
Lesson 9.1 Solving Quadratic Equations 1. Sketch the graph of a quadratic equation with a. One intercept and all nonnegative yvalues. b. The verte in the third quadrant and no intercepts. c. The verte
More information1.1 Practice Worksheet
Math 1 MPS Instructor: Cheryl Jaeger Balm 1 1.1 Practice Worksheet 1. Write each English phrase as a mathematical expression. (a) Three less than twice a number (b) Four more than half of a number (c)
More informationFlorida Algebra 1 EndofCourse Assessment Item Bank, Polk County School District
Benchmark: MA.912.A.2.3; Describe the concept of a function, use function notation, determine whether a given relation is a function, and link equations to functions. Also assesses MA.912.A.2.13; Solve
More informationALGEBRA I (Created 2014) Amherst County Public Schools
ALGEBRA I (Created 2014) Amherst County Public Schools The 2009 Mathematics Standards of Learning Curriculum Framework is a companion document to the 2009 Mathematics Standards of Learning and amplifies
More informationFactors and Products
CHAPTER 3 Factors and Products What You ll Learn use different strategies to find factors and multiples of whole numbers identify prime factors and write the prime factorization of a number find square
More informationAlgebra II A Final Exam
Algebra II A Final Exam Multiple Choice Identify the choice that best completes the statement or answers the question. Evaluate the expression for the given value of the variable(s). 1. ; x = 4 a. 34 b.
More information3. Power of a Product: Separate letters, distribute to the exponents and the bases
Chapter 5 : Polynomials and Polynomial Functions 5.1 Properties of Exponents Rules: 1. Product of Powers: Add the exponents, base stays the same 2. Power of Power: Multiply exponents, bases stay the same
More informationLearning Objectives 9.2. Media Run Times 9.3
Unit 9 Table of Contents Unit 9: Factoring Video Overview Learning Objectives 9.2 Media Run Times 9.3 Instructor Notes 9.4 The Mathematics of Factoring Polynomials Teaching Tips: Conceptual Challenges
More informationa) x 2 8x = 25 x 2 8x + 16 = (x 4) 2 = 41 x = 4 ± 41 x + 1 = ± 6 e) x 2 = 5 c) 2x 2 + 2x 7 = 0 2x 2 + 2x = 7 x 2 + x = 7 2
Solving Quadratic Equations By Square Root Method Solving Quadratic Equations By Completing The Square Consider the equation x = a, which we now solve: x = a x a = 0 (x a)(x + a) = 0 x a = 0 x + a = 0
More informationFactoring Polynomials and Solving Quadratic Equations
Factoring Polynomials and Solving Quadratic Equations Math Tutorial Lab Special Topic Factoring Factoring Binomials Remember that a binomial is just a polynomial with two terms. Some examples include 2x+3
More informationChapter 8. Chapter 8 Opener. Section 8.1. Big Ideas Math Green WorkedOut Solutions. Try It Yourself (p. 353) Number of cubes: 7
Chapter 8 Opener Try It Yourself (p. 5). The figure is a square.. The figure is a rectangle.. The figure is a trapezoid. g. Number cubes: 7. a. Sample answer: 4. There are 5 6 0 unit cubes in each layer.
More informationPark Forest Math Team. Meet #5. Algebra. Selfstudy Packet
Park Forest Math Team Meet #5 Selfstudy Packet Problem Categories for this Meet: 1. Mystery: Problem solving 2. Geometry: Angle measures in plane figures including supplements and complements 3. Number
More information3.3. The Factor Theorem. Investigate Determining the Factors of a Polynomial. Reflect and Respond
3.3 The Factor Theorem Focus on... factoring polynomials explaining the relationship between the linear factors of a polynomial expression and the zeros of the corresponding function modelling and solving
More informationIn this section, you will develop a method to change a quadratic equation written as a sum into its product form (also called its factored form).
CHAPTER 8 In Chapter 4, you used a web to organize the connections you found between each of the different representations of lines. These connections enabled you to use any representation (such as a graph,
More informationThe University of the State of New York REGENTS HIGH SCHOOL EXAMINATION INTEGRATED ALGEBRA. Tuesday, January 27, 2015 1:15 to 4:15 p.m.
INTEGRATED ALGEBRA The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION INTEGRATED ALGEBRA Tuesday, January 27, 2015 1:15 to 4:15 p.m., only Student Name: School Name: The possession
More informationSolve Absolute Value Equations and Inequalities
1.7 Solve Absolute Value Equations and Inequalities Before You solved linear equations and inequalities. Now You will solve absolute value equations and inequalities. Why? So you can describe hearing ranges
More informationUnit 1: Polynomials. Expressions:  mathematical sentences with no equal sign. Example: 3x + 2
Pure Math 0 Notes Unit : Polynomials Unit : Polynomials : Reviewing Polynomials Epressions:  mathematical sentences with no equal sign. Eample: Equations:  mathematical sentences that are equated with
More informationFactoring Guidelines. Greatest Common Factor Two Terms Three Terms Four Terms. 2008 Shirley Radai
Factoring Guidelines Greatest Common Factor Two Terms Three Terms Four Terms 008 Shirley Radai Greatest Common Factor 008 Shirley Radai Factoring by Finding the Greatest Common Factor Always check for
More informationIn algebra, factor by rewriting a polynomial as a product of lowerdegree polynomials
Algebra 2 Notes SOL AII.1 Factoring Polynomials Mrs. Grieser Name: Date: Block: Factoring Review Factor: rewrite a number or expression as a product of primes; e.g. 6 = 2 3 In algebra, factor by rewriting
More informationHigher Education Math Placement
Higher Education Math Placement Placement Assessment Problem Types 1. Whole Numbers, Fractions, and Decimals 1.1 Operations with Whole Numbers Addition with carry Subtraction with borrowing Multiplication
More informationAlgebra 2 PreAP. Name Period
Algebra 2 PreAP Name Period IMPORTANT INSTRUCTIONS FOR STUDENTS!!! We understand that students come to Algebra II with different strengths and needs. For this reason, students have options for completing
More informationAlgebra 1 Chapter 3 Vocabulary. equivalent  Equations with the same solutions as the original equation are called.
Chapter 3 Vocabulary equivalent  Equations with the same solutions as the original equation are called. formula  An algebraic equation that relates two or more reallife quantities. unit rate  A rate
More information( ) FACTORING. x In this polynomial the only variable in common to all is x.
FACTORING Factoring is similar to breaking up a number into its multiples. For example, 10=5*. The multiples are 5 and. In a polynomial it is the same way, however, the procedure is somewhat more complicated
More informationP.E.R.T. Math Study Guide
A guide to help you prepare for the Math subtest of Florida s Postsecondary Education Readiness Test or P.E.R.T. P.E.R.T. Math Study Guide www.perttest.com PERT  A Math Study Guide 1. Linear Equations
More informationFACTORING OUT COMMON FACTORS
278 (6 2) Chapter 6 Factoring 6.1 FACTORING OUT COMMON FACTORS In this section Prime Factorization of Integers Greatest Common Factor Finding the Greatest Common Factor for Monomials Factoring Out the
More informationMathematics Placement
Mathematics Placement The ACT COMPASS math test is a selfadaptive test, which potentially tests students within four different levels of math including prealgebra, algebra, college algebra, and trigonometry.
More informationThis is Factoring and Solving by Factoring, chapter 6 from the book Beginning Algebra (index.html) (v. 1.0).
This is Factoring and Solving by Factoring, chapter 6 from the book Beginning Algebra (index.html) (v. 1.0). This book is licensed under a Creative Commons byncsa 3.0 (http://creativecommons.org/licenses/byncsa/
More informationAlgebra 2 Unit 10 Tentative Syllabus Cubics & Factoring
Name Algebra Unit 10 Tentative Sllabus Cubics & Factoring DATE CLASS ASSIGNMENT Tuesda Da 1: S.1 Eponent s P: 1, 7 Jan Wednesda Da : S.1 More Eponent s P: 9 Jan Thursda Da : Graphing the cubic parent
More informationAlum Rock Elementary Union School District Algebra I Study Guide for Benchmark III
Alum Rock Elementary Union School District Algebra I Study Guide for Benchmark III Name Date Adding and Subtracting Polynomials Algebra Standard 10.0 A polynomial is a sum of one ore more monomials. Polynomial
More informationVocabulary Words and Definitions for Algebra
Name: Period: Vocabulary Words and s for Algebra Absolute Value Additive Inverse Algebraic Expression Ascending Order Associative Property Axis of Symmetry Base Binomial Coefficient Combine Like Terms
More informationAIP Factoring Practice/Help
The following pages include many problems to practice factoring skills. There are also several activities with examples to help you with factoring if you feel like you are not proficient with it. There
More informationPolynomials. Key Terms. quadratic equation parabola conjugates trinomial. polynomial coefficient degree monomial binomial GCF
Polynomials 5 5.1 Addition and Subtraction of Polynomials and Polynomial Functions 5.2 Multiplication of Polynomials 5.3 Division of Polynomials Problem Recognition Exercises Operations on Polynomials
More informationAlgebra I. In this technological age, mathematics is more important than ever. When students
In this technological age, mathematics is more important than ever. When students leave school, they are more and more likely to use mathematics in their work and everyday lives operating computer equipment,
More informationSECTION 16 Quadratic Equations and Applications
58 Equations and Inequalities Supply the reasons in the proofs for the theorems stated in Problems 65 and 66. 65. Theorem: The complex numbers are commutative under addition. Proof: Let a bi and c di be
More information13. Write the decimal approximation of 9,000,001 9,000,000, rounded to three significant
æ If 3 + 4 = x, then x = 2 gold bar is a rectangular solid measuring 2 3 4 It is melted down, and three equal cubes are constructed from this gold What is the length of a side of each cube? 3 What is the
More informationPreAlgebra Interactive Chalkboard Copyright by The McGrawHill Companies, Inc. Send all inquiries to:
PreAlgebra Interactive Chalkboard Copyright by The McGrawHill Companies, Inc. Send all inquiries to: GLENCOE DIVISION Glencoe/McGrawHill 8787 Orion Place Columbus, Ohio 43240 Click the mouse button
More informationWarmUp Oct. 22. Daily Agenda:
Evaluate y = 2x 3x + 5 when x = 1, 0, and 2. Daily Agenda: Grade Assignment Go over Ch 3 Test; Retakes must be done by next Tuesday 5.1 notes / assignment Graphing Quadratic Functions 5.2 notes / assignment
More informationOperations with Algebraic Expressions: Multiplication of Polynomials
Operations with Algebraic Expressions: Multiplication of Polynomials The product of a monomial x monomial To multiply a monomial times a monomial, multiply the coefficients and add the on powers with the
More informationMATH 21. College Algebra 1 Lecture Notes
MATH 21 College Algebra 1 Lecture Notes MATH 21 3.6 Factoring Review College Algebra 1 Factoring and Foiling 1. (a + b) 2 = a 2 + 2ab + b 2. 2. (a b) 2 = a 2 2ab + b 2. 3. (a + b)(a b) = a 2 b 2. 4. (a
More informationA. Factoring out the Greatest Common Factor.
DETAILED SOLUTIONS AND CONCEPTS  FACTORING POLYNOMIAL EXPRESSIONS Prepared by Ingrid Stewart, Ph.D., College of Southern Nevada Please Send Questions and Comments to ingrid.stewart@csn.edu. Thank you!
More informationModuMath Algebra Lessons
ModuMath Algebra Lessons Program Title 1 Getting Acquainted With Algebra 2 Order of Operations 3 Adding & Subtracting Algebraic Expressions 4 Multiplying Polynomials 5 Laws of Algebra 6 Solving Equations
More informationActually, if you have a graphing calculator this technique can be used to find solutions to any equation, not just quadratics. All you need to do is
QUADRATIC EQUATIONS Definition ax 2 + bx + c = 0 a, b, c are constants (generally integers) Roots Synonyms: Solutions or Zeros Can have 0, 1, or 2 real roots Consider the graph of quadratic equations.
More informationZeros of Polynomial Functions
Zeros of Polynomial Functions Objectives: 1.Use the Fundamental Theorem of Algebra to determine the number of zeros of polynomial functions 2.Find rational zeros of polynomial functions 3.Find conjugate
More informationFACTORING QUADRATICS 8.1.1 and 8.1.2
FACTORING QUADRATICS 8.1.1 and 8.1.2 Chapter 8 introduces students to quadratic equations. These equations can be written in the form of y = ax 2 + bx + c and, when graphed, produce a curve called a parabola.
More information