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Algebra 1 Vocabulary- Polynomial and Quadratics Unit Name_____________________________________ Date _____________________________________ Objective: To know the definitions of the key vocabulary words/terms/phrases in the Functions Unit. To understand the difference between an object and a process as they apply to the Functions Unit. In General Object As it applies to mathematical concepts In General Process As it applies to mathematical concepts 1. Classify each of the following words as an object and/or the result of a process. 2. What should be the expectation of the term as an object and/or the result of a process. Example: Root of a quadratic: Noun- The points at which a function crosses the x-axis Verb- The solution of a quadratic equation when set equal to 0. Graph Axes x-intercept y-intercept Coordinates Abscissa 1 Ordinate Independent variable Dependent variable Slope Symmetry Axis of Symmetry Operation Parabola Vertex Polynomial 2 Algebra 1 Lesson- Polynomial Operations- Addition & Subtraction Name:____________________________________ Date:_____________________________________ Objective: 1) 2) To review properties of monomials and polynomials To review the rules and processes for adding and subtracting monomials and polynomials DO NOW: Find the sum of 13, 25, and 32 without the use of a calculator __________________________________________________________________________________________ What is an operation? What is a monomial? What is a polynomial? What is the degree of a monomial? Situation 1: Monomial and Monomial (M&M) Rules: Examples: 3m 2 17m 2 1. 2. 9 z 3 12 z 3 3 Situation 2: Monomial and Binomial Rules: Examples: 3x (5 x 2) 3. 4. 2a (7a 1) Situation 3: Binomial and Binomial Rules: Examples: 5. (3x 2 2) (4 x 7) 6. (14 x 2 7) ( x 7) Situation 4: Polynomial and Polynomial Rules: Examples: 7. (3x 2 2 x 1) (4 x 7) 8. (14 x 2 7 x) ( x 2 7 x 6) 4 Algebra 1 Lesson- Polynomial Ops- Multiplication & Division Name:____________________________________ Date:_____________________________________ Objective: To review the rules and processes for multiplying and dividing monomials and polynomials DO NOW: Find the difference of 25 and 13 without the use of a calculator __________________________________________________________________________________________ Situation 1: Monomial and Monomial (M&M) Rules: Examples: 4m 2 6m 1. 2. 4m 2 6m 4. (2 x 2 4) (2 x) 6. ( x 4)( x 4) Situation 2: Monomial and Binomial Rules: Examples: 4 f (3a 2) 3. Situation 3: Multiplying Binomial by Binomial Rules: Examples: (2 x 1)( x 2) 5. 7. (2 x 1)( x 2 2 x 1) 5 Algebra 1 WKST- Polynomial Operations- Mixed Name:____________________________________ Date:_____________________________________ Perform the indicated operations: 1. 2 x 5 7 x 5 2. 16 x 3 7 x 3 5. 2 x 5 7 x 2 6. 21x 3 7 x 2 9. 4 x 2 (5 x 2 1) 10. 21x 3 (7 x 5x 3 ) 3. 5 x10 7 x10 7. 12 x 5 6 x 5 11. 2 x 5 (3 x 2 ) 4. 14 x 2 7 x 3 8. 24 y 5 9 x 5 12. (5 y 5 5) 5 y 13. (7 x 2 5) (3x 2 8) 14. (8x 3 4) (5 x 3 1) 15. (2 x 4)(3x 2) 16. (5 y 5 5) (5 y 3 2) 17. (7 x 2 4 x 12) (5x 2 11) 18. (2 x 3 4 x 1) (3x 2) 6 Algebra 1 Review- Polynomial Operations Quiz Name:____________________________________ Date:_____________________________________ Objective: To review the material that you will be tested on as part of a quiz. These topics are in the outline below: I. Working with Monomials a. Addition, subtraction, multiplying and dividing monomials b. Degree II. Operations with Polynomials a. Degree b. Adding Polynomials c. Subtracting Polynomials d. Multiplying Polynomials e. Dividing Polynomials __________________________________________________________________________________________ Below you will find a sample f the types of problems you can expect to see on the test. I.a. Perform the four basic operations ( ,,, ) on the following sets of monomials. 1. 6a, 12a 2. 3 xy, 6 xy 3. 25 x 2 yz 2 , 5 x 2 yz Multiply the following Monomials 4. (3a 2 ) 3 , 4a I.b. 5. 34 xyz, 10 x 3 yz 5 Determine the value of the coefficients and the degrees of the following monomials. 6. 2a 2 7. 5 8. 4xy 2 7 II.a. Determine the degree of the following polynomials. 9. 3 x 2 6 x 9 10. 14 xyz 2 x 5 II.b&c. Combine and simplify the following. 12. 3c 7d (4c 3d ) 5c 8d (d c) II.d. II.e. 11. 1 x 13. Subtract 3 p 2 9 p 7 from 2 p 2 7 Multiply the following polynomials and simplify. 14. 3c(4c 3d ) 15. ( x 2 1)(3x 2) 16. ( x 2)(2 x 2 x 4) Divide the following polynomials and simplify. 17. Divide 3x 6 x 12 by 3 2 6 s 3 8s 2 2 s 18. 2s 19. (30 m m 2 ) (5 m) 8 9 Algebra 1 Lesson: Factoring Out a GCF Name:____________________________________ Date:_____________________________________ Objective: To learn to factor out a Greatest Common Factor. Do Now: Find the prime factorization of 242 __________________________________________________________________________________________ What is a Factor? What is a GCF? How do you factor out a GCF? __________________________________________________________________________________________ Examples: 1. 5x2 5x 2. 10 xy 15x 2 y 2 3. c 3 c 2 2c 10 Algebra 1 Lesson- Factoring Perfect Square Trinomials Name_____________________________________ Date _____________________________________ Objective: To discover patterns within perfect square trinomials. Factor the following, noticing any patterns that arise. Use the space at the right to record your observations. 1. x 2 8 x 16 ( )( ) 2. x 2 16 x 64 ( )( ) 3. x 2 12 x 36 ( )( ) 4. x 2 10 x 25 ( )( ) Notes: Use the patterns you recognized in the problems above to factor the more difficult problems below. Define and then summarize the procedure for factoring perfect square trinomials in the right margin. 5. 16 x 2 8 x 1 ( )( ) 6. 9x 2 6x 1 ( )( ) 7. 4 x 2 20 x 25 ( )( ) 8. 16 x 2 24 x 9 ( )( ) 9. 9 x 2 30 x 25 ( )( ) Fill in the blanks so that each expression is a perfect square trinomial, then factor. 10. x 2 18x __________ ( )( )( )2 11. x 2 __________ 121 ( )( )( )2 12. 9 x 2 __________ 36 ( )( )( )2 13. ________ x 2 20 x 25 ( )( )( )2 14. 49 x 2 28x ________ ( )( )( )2 11 Algebra 1 Lesson: Difference of Two Squares Name:____________________________________ Date:_____________________________________ Objective: Recognize difference of two squares in binomial form Factor difference of two squares DO NOW: Factor 16 x 2 4 x __________________________________________________________________________________________ What is a difference of two squares? How do you factor it? Examples: Multiply: ( x 2)( x 2) 1. 4. (1 x)( x 1) 2. (3 x)(3 x) 3. (3x 4)(3x 4) 5. (6 x 13)(6 x 13) 6. ( 3 x)( 3 x) Factor: 1. x2 1 2. 4x2 1 5. 16 x 4 9 6. 1 4 x 1 16 3. 9 y 2 16 4. 1 2 16 x 4 25 12 Algebra 1 Lesson: Factoring Trinomials Name:____________________________________ Date:_____________________________________ Objective: Recognize trinomials that are not differences of two squares or perfect square trinomials Factor any trinomial with a leading coefficient of 1. DO NOW: Factor 4 x 2 1 __________________________________________________________________________________________ Polynomials: There are 3 types of polynomials that we will focus on: 1. 2. 3. DOTS Perfect Squares Everything else Both a DOTS and a Perfect Square Trinomial can be identified by the fact that the 1st and last terms are always perfect squares. If there are only 2 terms and a – separates those terms, then it is DOTS. If there are 3 terms and the 1st and last are perfect squares, then it is a perfect square trinomial. But what if it is neither…? To Factor any trinomial, you must know the following cases: Case 1: (+ +) x 2 5 x 6 Case 2: (– +) n 2 7 n 12 Case 3: (+ –) y 2 5 y 6 Case 4: (– –) x 2 2 x 35 13 Algebra 1 Lesson: Factoring Completely Name:____________________________________ Date:_____________________________________ Objective: To be able to factor any polynomial completely DO NOW: Factor x 2 7 x 12 __________________________________________________________________________________________ Process for Factoring Completely: (Usually Multiple Steps) Examples: 1. by 2 4b 3. 6x2 6 y 2 2. 3x 2 6 x 24 4. 10a 3 20a 2 10a 14 Algebra 1 WKST- Mixed Factoring Name_____________________________________ Date _____________________________________ Objective: To evaluate prior factoring skills Factor completely. If the polynomial can’t be factored, write prime. 1) x 2 64 2) x 2 11x 28 3) x 2 7 x 18 4) x 2 4 x 12 5) x 3 225 x 6) 7) x 2 6x 2 8) 9) x 2 3x 2 5 y 5 135 y 12 x 2 y 3 18 x 3 y 4 15 Algebra 1 Review- Factoring Name_____________________________________ Date _____________________________________ Objective: To Review Factoring of: Perfect square trinomials Difference of two squares Sum of two squares Factoring a Trinomial Perfect Square Trinomial To factor a perfect square trinomial follow the steps below: 1) Create two empty binomials as indicated on the right ( 2) Take the square root of the first term of the given trinomial 3) Take the square root of the last term of the given trinomial 4) Take the result of step 2 and put in the 1st position in each binomial 5) Take the result of step 3 and put in the 2nd position in each binomial 6) The signs in the binomials should be the same as the middle term of the binomial Example 1: Answer: )( ) factor : 4 x 2 12 x 9 (2x - 3)2 Difference of Two Squares a 2 b 2 To factor a difference of two squares follow the steps below: 1) Create two empty binomials as indicated on the right ( )( ) st 2) Take the square root of the first term of the given binomial and put in the 1 position of each binomial 3) Take the square root of the last term of the given binomial and put in the 2nd position of each binomial 4) Make one binomial a sum and the other binomial a difference Example 2: Answer: 9 x 2 y 2 16 (3xy - 4) (3xy + 4) a 2 b2 Sum of Two Squares The sum of squares is not factorable 16 Sum of Two Cubes To factor a sum of two cubes follow the steps below: 1) Create an empty binomial and an empty trinomial as indicated on the right ( )( 2) Take the cube root of the first term of the given expression and put it in the 1st position in the binomial. Square it and put it in the first position of the trinomial. **note- ignore all signs until the last step** 3) Take the cube root of the last term of the given expression and a. put it in the 2nd position in the binomial b. square it and put it in the last position of the trinomial 4) Find the product of the terms in the binomial and put it in the middle position of the trinomial 5) Arrange the signs as follows ( + )( − + ) ) Factoring a Trinomial Steps: 1) Check for a GCF. If one can be removed, remove it. 2) Determine the values that multiply to the “last term” that also add to the “middle” term. Signed numbers should be used here. Example 6: x 2 7 x 18 ( x 9)( x 2) 17 Algebra 1 Review- Mixed Polynomials and Factoring Name:____________________________________ Date:_____________________________________ Answer each of the following neatly and completely in the space provided. 1. Multiply: 3x 2 y 2 x 3 y 2 2. Multiply: 3x(2 x 6) 3. Multiply: (3x 2)( 2 x 1) 4. Multiply: x( x 1)( 2 x 2) 5. Add: 3x 2 2 xy 6. Add: 3x (2 x 6) 7. Add: (3x 2) (2 x 1) 8. Add: x ( x 1) (2 x 2) 9. Subtract: 3x 2 2 xy 10. Subtract: 3x (2 x 6) 11. Subtract: (3x 2) (2 x 1) 12. Subtract: x ( x 1) (2 x 2) 13. Factor: x 2 10 x 14. Factor: 25 x 2 16 15. Factor: a 2 4a 21 16. Factor: 17. Multiply: ( x 2)( x 2) 18. Multiply: (3 x) 2 19. Multiply: (3x 4)(3x 4) 20. Multiply: (1 x)( x 1) 21. Multiply: (6 x 13) 2 22. Multiply: ( 4 x)( 4 x) 23. Factor: x2 2x 1 24. Factor: x2 6x 9 25. Factor: 9 y 2 24 y 16 26. Factor: 16 x 4 24 x 2 9 x 2 10 x 25 18 Algebra 1 Lesson- Graphing Quadratic Functions Name:____________________________________ Date:_____________________________________ Objective: Do Now: To graph a quadratic function using the Roots Axis of symmetry Vertex y Graph the following function. f ( x) 3x 1 x Roots: Axis of symmetry: Vertex: __________________________________________________________________________________________ Graph each of the following on the same set of axes by finding the intercepts, axis of symmetry and vertex. 1. y x 2 4 x 2. y x 2 2 x 5 3. y x 2 4 x 4 Based on the graphs above, is there a shortcut for determining if the parabola opens up or down? 19 20 Algebra 1 Lesson- Zero Product Rule Name:____________________________________ Date:_____________________________________ Objective: To apply the zero product method for finding the roots of a quadratic function. Do Now: Create a table of values to graph the following function: y x 2 14 x 1 Zero product rule: Find the roots of the given quadratic using the zero product method: 1. x 2 4 x 3 0 2. x 2 7 x 6 0 3. x 2 10 x 0 4. 3 x 2 x 5. x( x 4) 5 6. x 2 2( x 12) 7. x 3 2 x 1 9. x 2 25 0 8. 3x x 2 2 4 10. 16 x 2 64 21 Algebra 1 HW- Solving for the Roots of Quadratics Name:____________________________________ Date:_____________________________________ Answer each of the following neatly and completely. Find the roots for each of the following equations using both an a graphing and algebraic approach. 1. x 2 81 2. s2 s 0 3. y2 6 y 4. 1 2 7 x x0 2 6 5. z 2 5z 4 0 6. x 2 11x 24 0 7. x 2 8 x 15 0 8. y2 y 0 9. x 27 3 x 10. 4 x 2 36 0 11. x 2 25 12. x 2 3 x 10 0 22 Algebra 1 Lesson: Quadratic Word Problems Name:____________________________________ Date:_____________________________________ Objective: To learn to interpret quadratic word problems, write symbolically and solve. Do Now: Solve for x: x x2 x 1 2 __________________________________________________________________________________________ Method: __________________________________________________________________________________________ 1 You have to make a square-bottomed, unlidded box with a height of three inches and a volume of approximately 42 cubic inches. You will be taking a piece of cardboard, cutting three-inch squares from each corner, scoring between the corners, and folding up the edges. What should be the dimensions of the cardboard, to the nearest quarter inch? 2. The product of two consecutive negative integers is 1122. What are the numbers? 3. You have a 500-foot roll of fencing and a large field. You want to construct a rectangular playground area. What are the dimensions of the largest such yard? What is the largest area? 23 4. In right triangle ABC, AB is two more than BC, AC is one more than BC. Find the length of each side of right triangle ABC. 5. The product of two consecutive ODD integers is 99. Find the integers. 6. The length of a rectangular garden is 4 meters more than the width. The area of the garden is 60 square meters. Find the dimensions of the garden. 7. a. b. c. If the length of a side of a square is represented by x + 6, express its area in terms of x. If the dimensions of a rectangle are represented by 3x + 2 & x – 2, express its area in terms of x. If the area of the square is equal to the area of the rectangle, find x and the value of the dimensions of each shape. 24 Algebra 1 Name:________________________________________ Wkst- Quadratic Word Problems Date:_____________________________________ A rectangular dog pen is to be made along an existing fence. The total length of the new fence is to be 100 feet and the width is x. Existing fence x pen x a) Express, in simplest from, the area of the pen in terms of x. b) What is the width that gives the maximum area? Show or explain how you arrived at your answer. c) What is the maximum area? 25 Another version of the fence problem… The owner takes down the fence and now wants to use the 100 feet of fencing to make two adjacent dog pens against the existing wall. See the diagram at the right. Existing wall x x x a) If x is the length of fence perpendicular to the existing wall, express, as a function of x, the length of the fence parallel to the wall. b) Express, as a function of x, the total area of both pens. c) What is the value of x (to the nearest tenth) that gives the maximum area? Show or explain how you arrived at your answer. d) What is the maximum total area(to the nearest whole unit)? 26 Algebra 1 Lesson- Solving quadratic linear systems Name:____________________________________ Date:_____________________________________ Objective: To learn how to solve systems of equations involving a line and a parabola. Do Now: Solve for x: 2 x 2 3x 9 0 __________________________________________________________________________________________ Solving a quadratic linear system Graphical Process: Algebraic Process: Examples: y x 2 2x 3 1. 2x y 2 2. y x 2 2x 1 y 2x 3 3. y x2 1 y x 1 27 Algebra 1 Review- Quadratics Test Name:____________________________________ Date:_____________________________________ Objective: To review the following topics for a test on quadratics. I. Solving for the roots of quadratics a. by factoring b. by graphing calculator II. Graphing quadratics a. axis of symmetry b. vertex c. roots d. maximum/minimum III. Quadratic/Linear Systems a. Graphic Solution (including calculator) b. algebraic solution IV. Quadratic word problems Answer each of the following on separate paper. x 2 81 0 1. Solve by factoring: y 2 3 y 10 0 2. Solve by factoring: 3. Solve by factoring: 14 x x 2 49 4. Solve by GC: 3x 2 5 x 4 0 5. Write the equation of the axis of symmetry, tell whether the turning point is a maximum or minimum, find the coordinates of the vertex, find the roots (if they exist, and graph the parabola for: y x 2 10 x 6. Write the equation of the axis of symmetry, tell whether the turning point is a maximum or minimum, find the coordinates of the vertex, find the roots (if they exist, and graph the parabola for: 1 1 y x 2 3x 2 2 7. Write the equation of the axis of symmetry, tell whether the turning point is a maximum or minimum, find the coordinates of the vertex, find the roots (if they exist, and graph the parabola 1 for: y x 2 (4 x 4) 2 y x2 Find the solution of the following quadratic/linear system algebraically and graphically: y x2 8. y x2 3 9. Find the solution of the following quadratic/linear system algebraically and graphically: 10. Find the solution of the following quadratic/linear system algebraically and graphically: 11. In a rectangle, the measure of the height is 5 less than the measure of the base, and the area is 14. Write and solve an equation to find the dimensions of the rectangle. y x 2 y x2 1 y x3 28 12. The perimeter of a right triangle is 30 inches and the length of its hypotenuse is 13 inches. a. If x represents the length of one leg, represent the length of the other leg in terms of x. b. Write and solve an equation to find the lengths of the legs of the right triangle. 13. The perimeter of a rectangle is 27 cm and its area is 35 square cm. Find the length and width of the rectangle. y y x y x y x x 29