Factor and Solve Polynomial Equations. In Chapter 4, you learned how to factor the following types of quadratic expressions.
|
|
- Luke Pearson
- 6 years ago
- Views:
Transcription
1 TEKS 5.4 2A.1.A, 2A.2.A; P..A, P..B 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 Vocaulary factored completely factor y grouping quadratic form In Chapter 4, you learned how to factor the following types of quadratic expressions. Type Example General trinomial 2x 2 2 x (2x 1 5)(x 2 4) Perfect square trinomial x 2 1 8x (x 1 4) 2 Difference of two squares 9x (x 1 1)(x 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 e factored completely using techniques learned in Chapter 4. KEY CONCEPT For Your Noteook Factoring Polynomials Definition A factorale polynomial with integer coefficients is factored completely if it is written as a product of unfactorale polynomials with integer coefficients. Examples 2(x 1 1)(x 2 4) and 5x 2 (x 2 2 ) are factored completely. x(x 2 2 4) is not factored completely ecause x can e factored as (x 1 2)(x 2 2). E XAMPLE 1 Find a common monomial factor Factor the polynomial completely. a. x 1 2x x 5 x(x 2 1 2x 2 15) Factor common monomial. 5 x(x 1 5)(x 2 ) Factor trinomial.. 2y y 5 2y (y 2 2 9) Factor common monomial. 5 2y (y 1 )(y 2 ) Difference of two squares c. 4z z 1 16z 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 5
2 FACTORING PATTERNS In part () 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 cues. KEY CONCEPT For Your Noteook Special Factoring Patterns Sum of Two Cues Example a 1 5 (a 1 )(a 2 2 a 1 2 ) 8x (2x) 1 5 (2x 1 )(4x 2 2 6x 1 9) Difference of Two Cues Example a 2 5 (a 2 )(a 2 1 a 1 2 ) 64x (4x) (4x 2 1)(16x 2 1 4x 1 1) E XAMPLE 2 Factor the sum or difference of two cues Factor the polynomial completely. a. x x 1 4 Sum of two cues 5 (x 1 4)(x 2 2 4x 1 16). 16z z 2 5 2z 2 (8z 2 125) Factor common monomial. 5 2z 2 F (2z) 2 5 G Difference of two cues 5 2z 2 (2z 2 5)(4z z 1 25) GUIDED PRACTICE for Examples 1 and 2 Factor the polynomial completely. 1. x 2 7x x 2. y y w 2 27 FACTORING BY GROUPING For some polynomials, you can factor y grouping pairs of terms that have a common monomial factor. The pattern for factoring y grouping is shown elow. ra 1 r 1 sa 1 s 5 r(a 1 ) 1 s(a 1 ) 5 (r 1 s)(a 1 ) E XAMPLE Factor y grouping AVOID ERRORS An expression is not factored completely until all factors, such as x , cannot e factored further. Factor the polynomial x 2 x x 1 48 completely. x 2 x x x 2 (x 2 ) 2 16(x 2 ) Factor y grouping. 5 (x )(x 2 ) Distriutive property 5 (x 1 4)(x 2 4)(x 2 ) Difference of two squares 54 Chapter 5 Polynomials and Polynomial Functions
3 QUADRATIC FORM An expression of the form au 2 1 u 1 c, where u is any expression in x, is said to e in quadratic form. The factoring techniques you studied in Chapter 4 can sometimes e used to factor such expressions. E XAMPLE 4 Factor polynomials in quadratic form IDENTIFY QUADRATIC FORM The expression 16x is in quadratic form ecause it can e written as u where u 5 4x 2. Factor completely: (a) 16x and () 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 )(2x 2 ) Difference of two squares. 2p p p 2 5 2p 2 (p 6 1 5p 1 6) Factor common monomial. 5 2p 2 (p 1 )(p 1 2) Factor trinomial in quadratic form. GUIDED PRACTICE for Examples and 4 Factor the polynomial completely. 5. x 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 factorale quadratic equations. You can extend this technique to solve some higher-degree polynomial equations. E XAMPLE 5 TAKS PRACTICE: Multiple Choice What are the real-numer solutions of the equation 4x x 5 60x? A 0, 2,, 6 B 2, 0, C 0, Ï } 6, D 2, 2Ï } 6, 0, Ï } 6, Solution 4x x 5 60x Write original equation. AVOID ERRORS Do not divide each side of an equation y a variale or a variale expression, such as 4x. Doing so will result in the loss of solutions. 4x x 1 216x 5 0 Write in standard form. 4x(x x ) 5 0 Factor common monomial. 4x(x 2 2 9)(x 2 2 6) 5 0 Factor trinomial. 4 x(x 1 )(x 2 )(x 2 2 6) 5 0 Difference of two squares x 5 0, x 52, x 5, x 5 Ï } 6, or x 52Ï } 6 Zero product property c The correct answer is D. A B C D GUIDED PRACTICE for Example 5 Find the real-numer solutions of the equation. 8. 4x x 1 6x x x 5 14x x 1 15x 2 526x Factor and Solve Polynomial Equations 55
4 E XAMPLE 6 Solve a polynomial equation CITY PARK You are designing a marle asin that will hold a fountain for a city park. The asin s sides and ottom should e 1 foot thick. Its outer length should e twice its outer width and outer height. What should the outer dimensions of the asin e if it is to hold 6 cuic feet of water? ANOTHER WAY For alternative methods to solving the prolem in Example 6, turn to page 60 for the Prolem Solving Workshop. Solution Volume (cuic feet) 5 Interior length (feet) p Interior width (feet) p Interior height (feet) 6 5 (2x 2 2) p (x 2 2) p (x 2 1) 6 5 (2x 2 2)(x 2 2)(x 2 1) Write equation. 05 2x 2 8x x 2 40 Write in standard form. 05 2x 2 (x 2 4) 1 10(x 2 4) Factor y grouping. 05 (2x )(x 2 4) Distriutive property c The only real solution is x 5 4. The asin is 8 ft long, 4 ft wide, and 4 ft high. GUIDED PRACTICE for Example WHAT IF? In Example 6, what should the asin s dimensions e if it is to hold 128 cuic feet of water and have outer length 6x, width x, and height x? 5.4 EXERCISES SKILL PRACTICE HOMEWORK KEY 5 WORKED-OUT SOLUTIONS on p. WS1 for Exs. 7, 2, and 61 5 TAKS PRACTICE AND REASONING Exs. 9, 41, 6, 64, 66, and VOCABULARY The expression 8x x 2 is in? form ecause it can e written as 2u 2 1 5u 2 where u 5 2x. 2. WRITING What condition must the factorization of a polynomial satisfy in order for the polynomial to e factored completely? EXAMPLE 1 on p. 5 for Exs. 9 MONOMIAL FACTORS Factor the polynomial completely.. 14x x c 1 9c c 6. z 2 6z z 7. y y 8. 54m m 4 1 9m 9. TAKS REASONING What is the complete factorization of 2x 7 2 2x? A 2x (x 1 2)(x 2 2)(x 2 1 4) B 2x (x 2 1 2)(x 2 2 2) C 2x (x 2 1 4) 2 D 2x (x 1 2) 2 (x 2 2) 2 56 Chapter 5 Polynomials and Polynomial Functions
5 EXAMPLE 2 on p. 54 for Exs EXAMPLE on p. 54 for Exs EXAMPLE 4 on p. 55 for Exs EXAMPLE 5 on p. 55 for Exs SUM OR DIFFERENCE OF CUBES Factor the polynomial completely. 10. x y m n a c w z 1 20 FACTORING BY GROUPING Factor the polynomial completely. 18. x 1 x 2 1 x y 2 7y 2 1 4y n 1 5n 2 2 9n m 2 m 2 1 9m s 2 100s 2 2 s c 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 28. 6m m 4 1 m x x 2 108x ERROR ANALYSIS Descrie and correct the error in finding all real-numer solutions. 0. 8x (2x 1 )(4x 2 1 6x 1 9) 5 0 x 52 } 2 1. x 2 48x 5 0 x(x ) 5 0 x x 524 or x 5 4 SOLVING EQUATIONS Find the real-numer solutions of the equation. 2. y 2 5y s 5 50s 4. g 1 g 2 2 g m 1 6m 2 2 4m w w z z x 6 2 4x 4 2 9x p p 41. TAKS REASONING What are the real-numer solutions of the equation x x 2 1 9x 5 x? A 21, 0, B 2, 0, C 2, 0, } 1, D 2, 2} 1, 0, CHOOSING A METHOD Factor the polynomial completely using any method x 2 44x x 4. n 4 2 4n a 2 15a a c c 2 10c d 4 2 1d x x y 6 2 8y y z 5 2 z z 1 48 GEOMETRY Find the possile value(s) of x. 51. Area Volume Volume 5 125π x 2 4 2x 2 5 x 14 2x x x 1 2 x 2 1 CHOOSING A METHOD Factor the polynomial completely using any method. 54. x y ac 2 1 c 2 2 7ad 2 2 d x 2n 2 2x n CHALLENGE Factor a a a 4 2 2a 1 a 2 2 completely. 5.4 Factor and Solve Polynomial Equations 57
6 PROBLEM SOLVING EXAMPLE 6 on p. 56 for Exs ARCHAEOLOGY At the ruins of Caesarea, archaeologists discovered a huge hydraulic concrete lock with a volume of 945 cuic meters. The lock s dimensions are x meters high y 12x 2 15 meters long y 12x 2 21 meters wide. What is the height of the lock? 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 e in the ratio 1: : 6. What should the outer dimensions of the mold e if it is to hold 112 cuic centimeters of chocolate? 60. MULTI-STEP PROBLEM A production crew is assemling a three-level platform inside a stadium for a performance. The platform has the dimensions shown in the diagrams, and has a total volume of 1250 cuic 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?. Write an Equation Use what you know aout the total volume to write an equation involving x. c. Solve Solve the equation from part (). Use your solution to calculate the dimensions of each of the three levels of the platform. 61. SCULPTURE Suppose you have 250 cuic inches of clay with which to make a sculpture shaped as a rectangular prism. You want the height and width each to e 5 inches less than the length. What should the dimensions of the prism e? 62. MANUFACTURING A manufacturer wants to uild a rectangular stainless steel tank with a holding capacity of 670 gallons, or aout cuic feet. The tank s walls will e one half inch thick, and aout 6.42 cuic feet of steel will e used for the tank. The manufacturer wants the outer dimensions of the tank to e related as follows: The width should e 2 feet less than the length. The height should e 8 feet more than the length. What should the outer dimensions of the tank e? x x 1 8 x WORKED-OUT SOLUTIONS on p. WS1 5 TAKS PRACTICE AND REASONING
7 6. TAKS REASONING A platform shaped like a rectangular prism has dimensions x 2 2 feet y 2 2x feet y x 1 4 feet. Explain why the volume of the platform cannot e } 7 cuic feet. 64. TAKS REASONING In 2000 B.C., the Baylonians solved polynomial equations using tales of values. One such tale gave values of y 1 y 2. To e ale to use this tale, the Baylonians sometimes had to manipulate the equation, as shown elow. a x } ax 1 x 2 5 c 1 a2 x 2 } 2 5 a2 c } Original equation Multiply each side y a2 }. 1} ax 2 1 1} ax a2 c } Rewrite cues and squares. They then found a2 c } in the y 1 y 2 column of the tale. Because the corresponding y-value was y 5 } ax, they could conclude that x 5 } y. a a. Calculate y 1 y 2 for y 5 1, 2,,..., 10. Record the values in a tale.. Use your tale and the method descried aove to solve x 1 2x c. Use your tale and the method descried aove to solve x 1 2x d. How can you modify the method descried aove for equations of the form ax 4 1 x 5 c? 65. CHALLENGE Use the diagram to complete parts (a) (c). a. Explain why a 2 is equal to the sum of the volumes of solid I, solid II, and solid III.. Write an algeraic expression for the volume of each of the three solids. Leave your expressions in factored form. c. Use the results from parts (a) and () to derive the factoring pattern for a 2 given on page 54. II I a III a a REVIEW Lesson 2.1; TAKS Workook REVIEW TAKS Preparation p. 408; TAKS Workook MIXED REVIEW FOR TAKS 66. TAKS PRACTICE Which inequality est descries the range of the function represented y the graph shown? TAKS Oj. 2 A y B y C 2 y D 24 y TAKS PRACTICE A poster is shaped like an equilateral triangle with a side length of 0 inches. What is the approximate area of the poster? TAKS Oj. 8 F 195 in. 2 G 18 in. 2 H 90 in. 2 J 780 in. 2 TAKS PRACTICE at classzone.com x 22 2 y 0 in. EXTRA PRACTICE for Lesson 5.4, p ONLINE QUIZ at classzone.com 59
8 LESSON 5.4 TEKS a.5, a.6, 2A.2.A; P..B Using ALTERNATIVE METHODS Another Way to Solve Example 6, page 56 MULTIPLE REPRESENTATIONS In Example 6 on page 56, you solved a polynomial equation y factoring. You can also solve a polynomial equation using a tale or a graph. P ROBLEM CITY PARK You are designing a marle asin that will hold a fountain for a city park. The asin s sides and ottom should e 1 foot thick. Its outer length should e twice its outer width and outer height. What should the outer dimensions of the asin e if it is to hold 6 cuic feet of water? M ETHOD 1 Using a Tale One alternative approach is to write a function for the volume of the asin and make a tale of values for the function. Using the tale, you can find the value of x that makes the volume of the asin 6 cuic feet. STEP 1 Write the function. From the diagram, you can see that the volume y of water the asin can hold is given y this function: y5 (2x 2 2)(x 2 2)(x 2 1) STEP 2 Make a tale of values for the function. Use only positive values of x ecause the asin s dimensions must e positive. STEP Identify the value of x for which y 5 6. The tale shows that y 5 6 when x 5 4. X Y1= Y1 X Y1= Y1 c The volume of the asin is 6 cuic feet when x is 4 feet. So, the outer dimensions of the asin should e as follows: Length 5 2x 5 8 feet Width 5 x 5 4 feet Height 5 x 5 4 feet 60 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 asin 6 cuic feet. STEP 1 Write the function. From the diagram, you can see that the volume y of water the asin can hold is given y this function: y5 (2x 2 2)(x 2 2)(x 2 1) STEP 2 Graph the equations y 5 6 and y 5 (x 2 1)(2x 2 2)(x 2 2). Choose a viewing window that shows the intersection of the graphs. STEP Identify the coordinates of the intersection point. On a graphing calculator, you can use the intersect feature. The intersection point is (4, 6). Intersection X=4 Y=6 c The volume of the asin is 6 cuic feet when x is 4 feet. So, the outer dimensions of the asin should e 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 tale or using a graph. 1. x 1 4x 2 2 8x x 2 9x x x 2 11x 2 1 x x 4 1 x 2 15x 2 2 8x x 4 1 2x 1 6x x x 4 1 4x 1 8x 2 1 4x x x 1 29x x WHAT IF? In the prolem on page 60, suppose the asin is to hold 200 cuic feet of water. Find the outer dimensions of the asin using a tale and using a graph. 9. PACKAGING A factory needs a ox that has a volume of 1728 cuic inches. The width should e 4 inches less than the height, and the length should e 6 inches greater than the height. Find the dimensions of the ox using a tale 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 e modeled y the function P(x) x x x x where x is the numer of years since In what year was the pineapple consumption aout 9.97 pounds per person? Solve the prolem using a tale and a graph. Using Alternative Methods 61
Evaluate and Graph Polynomial Functions
5.2 Evaluate and Graph Polynomial Functions Before You evaluated and graphed linear and quadratic functions. Now You will evaluate and graph other polynomial functions. Why? So you can model skateboarding
More informationFind a common monomial factor. = 2y 3 (y + 3)(y 3) Difference of two squares
EXAMPLE 1 Find a common monomial factor Factor the polynomial completely. a. x 3 + 2x 2 15x = x(x 2 + 2x 15) Factor common monomial. = x(x + 5)(x 3) Factor trinomial. b. 2y 5 18y 3 = 2y 3 (y 2 9) Factor
More informationSolving Polynomial Equations 3.5. Essential Question How can you determine whether a polynomial equation has a repeated solution?
3. Solving Polynomial Equations Essential Question Essential Question How can you determine whether a polynomial equation has a repeated solution? Cubic Equations and Repeated Solutions USING TOOLS STRATEGICALLY
More informationNC Math 3 Modelling with Polynomials
NC Math 3 Modelling with Polynomials Introduction to Polynomials; Polynomial Graphs and Key Features Polynomial Vocabulary Review Expression: Equation: Terms: o Monomial, Binomial, Trinomial, Polynomial
More informationSolve Radical Equations
6.6 Solve Radical Equations TEKS 2A.9.B, 2A.9.C, 2A.9.D, 2A.9.F Before Now You solved polynomial equations. You will solve radical equations. Why? So you can calculate hang time, as in Ex. 60. Key Vocabulary
More informationFor Your Notebook E XAMPLE 1. Factor when b and c are positive KEY CONCEPT. CHECK (x 1 9)(x 1 2) 5 x 2 1 2x 1 9x Factoring x 2 1 bx 1 c
9.5 Factor x2 1 bx 1 c Before You factored out the greatest common monomial factor. Now You will factor trinomials of the form x 2 1 bx 1 c. Why So you can find the dimensions of figures, as in Ex. 61.
More informationAdd, Subtract, and Multiply Polynomials
TEKS 5.3 a.2, 2A.2.A; P.3.A, P.3.B Add, Subtract, and Multiply Polynomials Before You evaluated and graphed polynomial functions. Now You will add, subtract, and multiply polynomials. Why? So you can model
More informationSolve Radical Equations
6.6 Solve Radical Equations Before You solved polynomial equations. Now You will solve radical equations. Why? So you can calculate hang time, as in Ex. 60. Key Vocabulary radical equation extraneous solution,
More informationSolve Trigonometric Equations. Solve a trigonometric equation
14.4 a.5, a.6, A..A; P.3.D TEKS Before Now Solve Trigonometric Equations You verified trigonometric identities. You will solve trigonometric equations. Why? So you can solve surface area problems, as in
More informationSolve Quadratic Equations by Completing the Square
10.5 Solve Quadratic Equations by Completing the Square Before You solved quadratic equations by finding square roots. Now You will solve quadratic equations by completing the square. Why? So you can solve
More informationQUADRATIC EQUATIONS EXPECTED BACKGROUND KNOWLEDGE
6 QUADRATIC EQUATIONS In this lesson, you will study aout quadratic equations. You will learn to identify quadratic equations from a collection of given equations and write them in standard form. You will
More informationFind Sums of Infinite Geometric Series
a, AA; PB, PD TEKS Find Sums of Infinite Geometric Series Before You found the sums of finite geometric series Now You will find the sums of infinite geometric series Why? So you can analyze a fractal,
More information3.5 Solving Quadratic Equations by the
www.ck1.org Chapter 3. Quadratic Equations and Quadratic Functions 3.5 Solving Quadratic Equations y the Quadratic Formula Learning ojectives Solve quadratic equations using the quadratic formula. Identify
More informationCompleting the Square
3.5 Completing the Square Essential Question How can you complete the square for a quadratic epression? Using Algera Tiles to Complete the Square Work with a partner. Use algera tiles to complete the square
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? Explore Analyzing a Visual Model for Polynomial Factorization Factoring
More informationPolynomial Degree and Finite Differences
CONDENSED LESSON 7.1 Polynomial Degree and Finite Differences In this lesson, you Learn the terminology associated with polynomials Use the finite differences method to determine the degree of a polynomial
More informationObjectives To solve equations by completing the square To rewrite functions by completing the square
4-6 Completing the Square Content Standard Reviews A.REI.4. Solve quadratic equations y... completing the square... Ojectives To solve equations y completing the square To rewrite functions y completing
More informationWarm Up Lesson Presentation Lesson Quiz. Holt McDougal Algebra 2
Warm Up Lesson Presentation Lesson Quiz Algebra 2 Warm Up Factor each expression. 1. 3x 6y 2. a 2 b 2 3(x 2y) (a + b)(a b) Find each product. 3. (x 1)(x + 3) 4. (a + 1)(a 2 + 1) x 2 + 2x 3 a 3 + a 2 +
More informationMaintaining Mathematical Proficiency
Chapter 7 Maintaining Mathematical Proficiency Simplify the expression. 1. 5x 6 + 3x. 3t + 7 3t 4 3. 8s 4 + 4s 6 5s 4. 9m + 3 + m 3 + 5m 5. 4 3p 7 3p 4 1 z 1 + 4 6. ( ) 7. 6( x + ) 4 8. 3( h + 4) 3( h
More informationFinding Complex Solutions of Quadratic Equations
y - y - - - x x Locker LESSON.3 Finding Complex Solutions of Quadratic Equations Texas Math Standards The student is expected to: A..F Solve quadratic and square root equations. Mathematical Processes
More informationA linear inequality in one variable can be written in one of the following forms, where a and b are real numbers and a Þ 0:
TEKS.6 a.2, a.5, A.7.A, A.7.B 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
More informationGraph and Write Equations of Ellipses. You graphed and wrote equations of parabolas and circles. You will graph and write equations of ellipses.
TEKS 9.4 a.5, A.5.B, A.5.C Before Now Graph and Write Equations of Ellipses You graphed and wrote equations of parabolas and circles. You will graph and write equations of ellipses. Wh? So ou can model
More informationWrite and Apply Exponential and Power Functions
TEKS 7.7 a., 2A..B, 2A..F Write and Apply Exponential and Power Functions Before You wrote linear, quadratic, and other polynomial functions. Now You will write exponential and power functions. Why? So
More informationSECTION 1.4 PolyNomiAls feet. Figure 1. A = s 2 = (2x) 2 = 4x 2 A = 2 (2x) 3 _ 2 = 1 _ = 3 _. A = lw = x 1. = x
SECTION 1.4 PolyNomiAls 4 1 learning ObjeCTIveS In this section, you will: Identify the degree and leading coefficient of polynomials. Add and subtract polynomials. Multiply polynomials. Use FOIL to multiply
More informationGraph Square Root and Cube Root Functions
TEKS 6.5 2A.4.B, 2A.9.A, 2A.9.B, 2A.9.F Graph Square Root and Cube Root Functions Before You graphed polnomial functions. Now You will graph square root and cube root functions. Wh? So ou can graph the
More informationUse direct substitution to evaluate the polynomial function for the given value of x
Checkpoint 1 Decide whether the function is a polynomial function. If so, write it in standard form and state its degree, type, and leading coefficient 1. f(x) = 8 x 2 2. f(x) = 6x + 8x 4 3 3. g(x) = πx
More informationACTIVITY: Factoring Special Products. Work with a partner. Six different algebra tiles are shown below.
7.9 Factoring Special Products special products? How can you recognize and factor 1 ACTIVITY: Factoring Special Products Work with a partner. Six different algebra tiles are shown below. 1 1 x x x 2 x
More information7.7. Factoring Special Products. Essential Question How can you recognize and factor special products?
7.7 Factoring Special Products Essential Question How can you recognize and factor special products? Factoring Special Products LOOKING FOR STRUCTURE To be proficient in math, you need to see complicated
More information21.1 Solving Equations by Factoring
Name Class Date 1.1 Solving Equations by Factoring x + bx + c Essential Question: How can you use factoring to solve quadratic equations in standard form for which a = 1? Resource Locker Explore 1 Using
More informationPerform Basic Matrix Operations
TEKS 3.5 a.1, a. Perform Basic Matrix Operations Before You performed operations with real numbers. Now You will perform operations with matrices. Why? So you can organize sports data, as in Ex. 34. Key
More information10.7. Interpret the Discriminant. For Your Notebook. x5 2b 6 Ï} b 2 2 4ac E XAMPLE 1. Use the discriminant KEY CONCEPT
10.7 Interpret the Discriminant Before You used the quadratic formula. Now You will use the value of the discriminant. Wh? So ou can solve a problem about gmnastics, as in E. 49. Ke Vocabular discriminant
More informationSolve Linear Systems Algebraically
TEKS 3.2 a.5, 2A.3.A, 2A.3.B, 2A.3.C Solve Linear Systems Algebraically Before You solved linear systems graphically. Now You will solve linear systems algebraically. Why? So you can model guitar sales,
More informationControlling the Population
Lesson.1 Skills Practice Name Date Controlling the Population Adding and Subtracting Polynomials Vocabulary Match each definition with its corresponding term. 1. polynomial a. a polynomial with only 1
More informationAnalyze Geometric Sequences and Series
23 a4, 2A2A; P4A, P4B TEKS Analyze Geometric Sequences and Series Before You studied arithmetic sequences and series Now You will study geometric sequences and series Why? So you can solve problems about
More informationWords Algebra Graph. 5 rise } run. } x2 2 x 1. m 5 y 2 2 y 1. slope. Find slope in real life
TEKS 2.2 a.1, a.4, a.5 Find Slope and Rate of Change Before You graphed linear functions. Now You will find slopes of lines and rates of change. Wh? So ou can model growth rates, as in E. 46. Ke Vocabular
More information6 p p } 5. x 26 x 5 x 3 5 x Product of powers property x4 y x 3 y 2 6
Chapter Polnomials and Polnomial Functions Copright Houghton Mifflin Harcourt Publishing Compan. All rights reserved. Prerequisite Skills for the chapter Polnomials and Polnomial Functions. and. 4. a b
More informationSimplifying a Rational Expression. Factor the numerator and the denominator. = 1(x 2 6)(x 2 1) Divide out the common factor x 6. Simplify.
- Plan Lesson Preview Check Skills You ll Need Factoring ± ± c Lesson -5: Eamples Eercises Etra Practice, p 70 Lesson Preview What You ll Learn BJECTIVE - To simplify rational epressions And Why To find
More information9-8 Completing the Square
In the previous lesson, you solved quadratic equations by isolating x 2 and then using square roots. This method works if the quadratic equation, when written in standard form, is a perfect square. When
More informationCompleting the Square
5-7 Completing the Square TEKS FOCUS TEKS (4)(F) Solve quadratic and square root equations. TEKS (1)(A) Apply mathematics to problems arising in everyday life, society, and the workplace. Additional TEKS
More informationSolve Systems of Linear Equations in Three Variables
TEKS 3.4 a.5, 2A.3.A, 2A.3.B, 2A.3.C Solve Systems of Linear Equations in Three Variables Before You solved systems of equations in two variables. Now You will solve systems of equations in three variables.
More informationMultiplying Polynomials. The rectangle shown at the right has a width of (x + 2) and a height of (2x + 1).
Page 1 of 6 10.2 Multiplying Polynomials What you should learn GOAL 1 Multiply two polynomials. GOAL 2 Use polynomial multiplication in real-life situations, such as calculating the area of a window in
More informationModel Direct Variation. You wrote and graphed linear equations. You will write and graph direct variation equations.
2.5 Model Direct Variation a.3, 2A.1.B, TEKS 2A.10.G Before Now You wrote and graphed linear equations. You will write and graph direct variation equations. Why? So you can model animal migration, as in
More information( ) Chapter 7 ( ) ( ) ( ) ( ) Exercise Set The greatest common factor is x + 3.
Chapter 7 Exercise Set 7.1 1. A prime number is an integer greater than 1 that has exactly two factors, itself and 1. 3. To factor an expression means to write the expression as the product of factors.
More informationFind two positive factors of 24 whose sum is 10. Make an organized list.
9.5 Study Guide For use with pages 582 589 GOAL Factor trinomials of the form x 2 1 bx 1 c. EXAMPLE 1 Factor when b and c are positive Factor x 2 1 10x 1 24. Find two positive factors of 24 whose sum is
More informationTEKS: 2A.10F. Terms. Functions Equations Inequalities Linear Domain Factor
POLYNOMIALS UNIT TEKS: A.10F Terms: Functions Equations Inequalities Linear Domain Factor Polynomials Monomial, Like Terms, binomials, leading coefficient, degree of polynomial, standard form, terms, Parent
More informationAdditional Exercises 7.1 Form I The Greatest Common Factor and Factoring by Grouping
Additional Exercises 7.1 Form I The Greatest Common Factor and Factoring by Grouping Find the greatest common factor of each list of monomials. 1. 10x and 15 x 1.. 3 1y and 8y. 3. 16 a 3 a, 4 and 4 3a
More informationSolve Absolute Value Equations and Inequalities
TEKS 1.7 a.1, a.2, a.5, 2A.2.A Solve Absolute Value Equations and Inequalities Before You solved linear equations and inequalities. Now You will solve absolute value equations and inequalities. Why? So
More informationChapter 5: Exponents and Polynomials
Chapter 5: Exponents and Polynomials 5.1 Multiplication with Exponents and Scientific Notation 5.2 Division with Exponents 5.3 Operations with Monomials 5.4 Addition and Subtraction of Polynomials 5.5
More informationGraph and Write Equations of Parabolas
TEKS 9.2 a.5, 2A.5.B, 2A.5.C Graph and Write Equations of Parabolas Before You graphed and wrote equations of parabolas that open up or down. Now You will graph and write equations of parabolas that open
More informationEvaluate and Simplify Algebraic Expressions
TEKS 1.2 a.1, a.2, 2A.2.A, A.4.B Evaluate and Simplify Algebraic Expressions Before You studied properties of real numbers. Now You will evaluate and simplify expressions involving real numbers. Why? So
More informationPrerequisites. Introduction CHAPTER OUTLINE
Prerequisites 1 Figure 1 Credit: Andreas Kambanls CHAPTER OUTLINE 1.1 Real Numbers: Algebra Essentials 1.2 Exponents and Scientific Notation 1.3 Radicals and Rational Expressions 1.4 Polynomials 1.5 Factoring
More informationApply Properties of Logarithms. Let b, m, and n be positive numbers such that b Þ 1. m 1 log b. mn 5 log b. m }n 5 log b. log b.
TEKS 7.5 a.2, 2A.2.A, 2A.11.C Apply Properties of Logarithms Before You evaluated logarithms. Now You will rewrite logarithmic epressions. Why? So you can model the loudness of sounds, as in E. 63. Key
More informationproportion, p. 163 cross product, p. 168 scale drawing, p. 170
REVIEW KEY VOCABULARY classzone.com Multi-Language Glossary Vocabulary practice inverse operations, p. 14 equivalent equations, p. 14 identity, p. 156 ratio, p. 162 proportion, p. 16 cross product, p.
More informationSolving Quadratic Equations
Solving Quadratic Equations MATH 101 College Algebra J. Robert Buchanan Department of Mathematics Summer 2012 Objectives In this lesson we will learn to: solve quadratic equations by factoring, solve quadratic
More informationSolve Exponential and Logarithmic Equations. You studied exponential and logarithmic functions. You will solve exponential and logarithmic equations.
TEKS 7.6 Solve Exponential and Logarithmic Equations 2A..A, 2A..C, 2A..D, 2A..F Before Now You studied exponential and logarithmic functions. You will solve exponential and logarithmic equations. Why?
More informationYou evaluated powers. You will simplify expressions involving powers. Consider what happens when you multiply two powers that have the same base:
TEKS.1 a.1, 2A.2.A Before Now Use Properties of Eponents You evaluated powers. You will simplify epressions involving powers. Why? So you can compare the volumes of two stars, as in Eample. Key Vocabulary
More informationWarm Up Lesson Presentation Lesson Quiz. Holt Algebra 2 2
6-5 Warm Up Lesson Presentation Lesson Quiz 2 Warm Up Factor completely. 1. 2y 3 + 4y 2 30y 2y(y 3)(y + 5) 2. 3x 4 6x 2 24 Solve each equation. 3(x 2)(x + 2)(x 2 + 2) 3. x 2 9 = 0 x = 3, 3 4. x 3 + 3x
More information4-9 Using Similar Figures. Warm Up Problem of the Day Lesson Presentation Lesson Quizzes
Warm Up Problem of the Day Lesson Presentation Lesson Quizzes Warm Up Solve each proportion. 1. k 4 = 75 25 3. k = 12 2. 6 19 = 24 x = 76 x Triangles JNZ and KOA are similar. Identify the side that corresponds
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 informationSolve Quadratic Equations by Graphing
0.3 Solve Quadratic Equations b Graphing Before You solved quadratic equations b factoring. Now You will solve quadratic equations b graphing. Wh? So ou can solve a problem about sports, as in Eample 6.
More informationApply Properties of 1.1 Real Numbers
TEKS Apply Properties of 1.1 Real Numbers a.1, a.6 Before Now You performed operations with real numbers. You will study properties of real numbers. Why? So you can order elevations, as in Ex. 58. Key
More informationMAFS Algebra 1. Polynomials. Day 15 - Student Packet
MAFS Algebra 1 Polynomials Day 15 - Student Packet Day 15: Polynomials MAFS.91.A-SSE.1., MAFS.91.A-SSE..3a,b, MAFS.91.A-APR..3, MAFS.91.F-IF.3.7c I CAN rewrite algebraic expressions in different equivalent
More information10-6 Changing Dimensions. IWBAT find the volume and surface area of similar three-dimensional figures.
IWBAT find the volume and surface area of similar three-dimensional figures. Recall that similar figures have proportional side lengths. The surface areas of similar three-dimensional figures are also
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 informationHow can you factor the trinomial x 2 + bx + c into the product of two binomials? ACTIVITY: Finding Binomial Factors
7.7 Factoring x 2 + bx + c How can you factor the trinomial x 2 + bx + c into the product of two binomials? 1 ACTIVITY: Finding Binomial Factors Work with a partner. Six different algebra tiles are shown
More informationQuick-and-Easy Factoring. of lower degree; several processes are available to fi nd factors.
Lesson 11-3 Quick-and-Easy Factoring BIG IDEA Some polynomials can be factored into polynomials of lower degree; several processes are available to fi nd factors. Vocabulary factoring a polynomial factored
More informationLESSON 10.1 QUADRATIC EQUATIONS I
LESSON 10.1 QUADRATIC EQUATIONS I LESSON 10.1 QUADRATIC EQUATIONS I 409 OVERVIEW Here s what you ll learn in this lesson: Solving by Factoring a. The standard form of a quadratic equation b. Putting a
More informationWrite Quadratic Functions and Models
4.0 A..B, A.6.B, A.6.C, A.8.A TEKS Write Quadratic Functions and Models Before You wrote linear functions and models. Now You will write quadratic functions and models. Wh? So ou can model the cross section
More informationDay 7: Polynomials MAFS.912.A-SSE.1.2, MAFS.912.A-SSE.2.3a,b, MAFS.912.A-APR.2.3, MAFS.912.F-IF.3.7c
Day 7: Polynomials MAFS.91.A-SSE.1., MAFS.91.A-SSE..3a,b, MAFS.91.A-APR..3, MAFS.91.F-IF.3.7c I CAN rewrite algebraic expressions in different equivalent forms using factoring techniques use equivalent
More informationRepresent Relations and Functions
TEKS. a., a., a.5, A..A Represent Relations and Functions Before You solved linear equations. Now You will represent relations and graph linear functions. Wh? So ou can model changes in elevation, as in
More informationLESSON 9.1 ROOTS AND RADICALS
LESSON 9.1 ROOTS AND RADICALS LESSON 9.1 ROOTS AND RADICALS 67 OVERVIEW Here s what you ll learn in this lesson: Square Roots and Cube Roots a. Definition of square root and cube root b. Radicand, radical
More informationFactoring x 2 + bx + c
7.5 Factoring x 2 + bx + c Essential Question How can you use algebra tiles to factor the trinomial x 2 + bx + c into the product of two binomials? Finding Binomial Factors Work with a partner. Use algebra
More informationMini-Lecture 5.1 Exponents and Scientific Notation
Mini-Lecture.1 Eponents and Scientific Notation Learning Objectives: 1. Use the product rule for eponents.. Evaluate epressions raised to the zero power.. Use the quotient rule for eponents.. Evaluate
More informationLESSON 7.2 FACTORING POLYNOMIALS II
LESSON 7.2 FACTORING POLYNOMIALS II LESSON 7.2 FACTORING POLYNOMIALS II 305 OVERVIEW Here s what you ll learn in this lesson: Trinomials I a. Factoring trinomials of the form x 2 + bx + c; x 2 + bxy +
More informationCommon Core Algebra 2. Chapter 3: Quadratic Equations & Complex Numbers
Common Core Algebra 2 Chapter 3: Quadratic Equations & Complex Numbers 1 Chapter Summary: The strategies presented for solving quadratic equations in this chapter were introduced at the end of Algebra.
More information5.1, 5.2, 5.3 Properites of Exponents last revised 6/7/2014. c = Properites of Exponents. *Simplify each of the following:
48 5.1, 5.2, 5.3 Properites of Exponents last revised 6/7/2014 Properites of Exponents 1. x a x b = x a+b *Simplify each of the following: a. x 4 x 8 = b. x 5 x 7 x = 2. xa xb = xa b c. 5 6 5 11 = d. x14
More informationa b a b ab b b b Math 154B Elementary Algebra Spring 2012
Math 154B Elementar Algera Spring 01 Stud Guide for Eam 4 Eam 4 is scheduled for Thursda, Ma rd. You ma use a " 5" note card (oth sides) and a scientific calculator. You are epected to know (or have written
More informationMathwithsheppard.weebly.com
Unit #: Powers and Polynomials Unit Outline: Date Lesson Title Assignment Completed.1 Introduction to Algebra. Discovering the Exponent Laws Part 1. Discovering the Exponent Laws Part. Multiplying and
More informationMaintaining Mathematical Proficiency
Chapter Maintaining Mathematical Proficiency Simplify the expression. 1. 8x 9x 2. 25r 5 7r r + 3. 3 ( 3x 5) + + x. 3y ( 2y 5) + 11 5. 3( h 7) 7( 10 h) 2 2 +. 5 8x + 5x + 8x Find the volume or surface area
More informationTest 4 also includes review problems from earlier sections so study test reviews 1, 2, and 3 also.
MATD 0370 ELEMENTARY ALGEBRA REVIEW FOR TEST 4 (1.1-10.1, not including 8.2) Test 4 also includes review problems from earlier sections so study test reviews 1, 2, and 3 also. 1. Factor completely: a 2
More informationSolving Systems of Linear Equations Symbolically
" Solving Systems of Linear Equations Symolically Every day of the year, thousands of airline flights crisscross the United States to connect large and small cities. Each flight follows a plan filed with
More informationAlgebra II Chapter 5: Polynomials and Polynomial Functions Part 1
Algebra II Chapter 5: Polynomials and Polynomial Functions Part 1 Chapter 5 Lesson 1 Use Properties of Exponents Vocabulary Learn these! Love these! Know these! 1 Example 1: Evaluate Numerical Expressions
More informationRational Expressions VOCABULARY
11-4 Rational Epressions TEKS FOCUS TEKS (7)(F) Determine the sum, difference, product, and quotient of rational epressions with integral eponents of degree one and of degree two. TEKS (1)(G) Display,
More informationUsing Properties of Exponents
6.1 Using Properties of Exponents Goals p Use properties of exponents to evaluate and simplify expressions involving powers. p Use exponents and scientific notation to solve real-life problems. VOCABULARY
More informationPolynomials 370 UNIT 10 WORKING WITH POLYNOMIALS. The railcars are linked together.
UNIT 10 Working with Polynomials The railcars are linked together. 370 UNIT 10 WORKING WITH POLYNOMIALS Just as a train is built from linking railcars together, a polynomial is built by bringing terms
More information5.3. Polynomials and Polynomial Functions
5.3 Polynomials and Polynomial Functions Polynomial Vocabulary Term a number or a product of a number and variables raised to powers Coefficient numerical factor of a term Constant term which is only a
More informationGraph and Write Equations of Circles
TEKS 9.3 a.5, A.5.B Graph and Write Equations of Circles Before You graphed and wrote equations of parabolas. Now You will graph and write equations of circles. Wh? So ou can model transmission ranges,
More informationName Class Date. Multiplying Two Binomials Using Algebra Tiles
Name Class Date Multiplying Polynomials Going Deeper Essential question: How do you multiply polynomials? 6-5 A monomial is a number, a variable, or the product of a number and one or more variables raised
More information( ) Chapter 6 ( ) ( ) ( ) ( ) Exercise Set The greatest common factor is x + 3.
Chapter 6 Exercise Set 6.1 1. A prime number is an integer greater than 1 that has exactly two factors, itself and 1. 3. To factor an expression means to write the expression as the product of factors.
More information= 9 = x + 8 = = -5x 19. For today: 2.5 (Review) and. 4.4a (also review) Objectives:
Math 65 / Notes & Practice #1 / 20 points / Due. / Name: Home Work Practice: Simplify the following expressions by reducing the fractions: 16 = 4 = 8xy =? = 9 40 32 38x 64 16 Solve the following equations
More informationAlgebra I. Exponents and Polynomials. Name
Algebra I Exponents and Polynomials Name 1 2 UNIT SELF-TEST QUESTIONS The Unit Organizer #6 2 LAST UNIT /Experience NAME 4 BIGGER PICTURE DATE Operations with Numbers and Variables 1 CURRENT CURRENT UNIT
More informationJakarta International School 8 th Grade AG1 Practice Test - BLACK
Jakarta International School 8 th Grade AG1 Practice Test - BLACK Polynomials and Quadratic Equations Name: Date: Grade: Standard Level Learning Goals - Green Understand and Operate with Polynomials Graph
More informationEvaluate Logarithms and Graph Logarithmic Functions
TEKS 7.4 2A.4.C, 2A..A, 2A..B, 2A..C Before Now Evaluate Logarithms and Graph Logarithmic Functions You evaluated and graphed eponential functions. You will evaluate logarithms and graph logarithmic functions.
More informationUse Scientific Notation
8.4 Use Scientific Notation Before You used properties of exponents. Now You will read and write numbers in scientific notation. Why? So you can compare lengths of insects, as in Ex. 51. Key Vocabulary
More informationEssential Question How can you factor a polynomial completely?
REASONING ABSTRACTLY 7.8 To be proficient in math, ou need to know and flexibl use different properties of operations and objects. Factoring Polnomials Completel Essential Question How can ou factor a
More informationYou studied exponential growth and decay functions.
TEKS 7. 2A.4.B, 2A..B, 2A..C, 2A..F Before Use Functions Involving e You studied eponential growth and deca functions. Now You will stud functions involving the natural base e. Wh? So ou can model visibilit
More informationMath Analysis CP WS 4.X- Section Review A
Math Analysis CP WS 4.X- Section 4.-4.4 Review Complete each question without the use of a graphing calculator.. Compare the meaning of the words: roots, zeros and factors.. Determine whether - is a root
More informationMath 3 Variable Manipulation Part 7 Absolute Value & Inequalities
Math 3 Variable Manipulation Part 7 Absolute Value & Inequalities 1 MATH 1 REVIEW SOLVING AN ABSOLUTE VALUE EQUATION Absolute value is a measure of distance; how far a number is from zero. In practice,
More informationSolve for the variable by transforming equations:
Cantwell Sacred Heart of Mary High School Math Department Study Guide for the Algebra 1 (or higher) Placement Test Name: Date: School: Solve for the variable by transforming equations: 1. y + 3 = 9. 1
More informationNAME DATE PERIOD. Study Guide and Intervention. Solving Polynomial Equations. For any number of terms, check for: greatest common factor
5-5 Factor Polynomials Study Guide and Intervention For any number of terms, check for: greatest common factor Techniques for Factoring Polynomials For two terms, check for: Difference of two squares a
More information