# ALGEBRA 1. Name: Unit 4 Chapter 8

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1 ALGEBRA 1 Name: Unit 4 Chapter 8 UPDATED SPRING

2 Section 8.1 Notes: Adding and Subtracting Polynomials Warm-Up Simplify. Assume that no denominator is equal to zero. 1) 70a2 b 3 c 14a 1 b 2 c 2) (-2x 4 y)(-3x 2 ) 2 Simplify. 3 3) 216 4) Polynomial: a monomial or the of monomials, each called a of the polynomial. Binomial: the sum of Trinomial: the sum of monomials. monomials. Degree of a monomial: the sum of the Degree of a polynomial: the of all its variables. A nonzero constant term has a degree of. degree of any term in the polynomial. Polynomials are named based on their degree. Example 1: State whether each expression is a polynomial. If it is a polynomial, find the degree and determine whether it is a monomial, binomial, or trinomial. Expression Is it a polynomial? Degree Monomial, binomial, or trinomial? a.) 4y 5xz Yes; 4y 5xz is the sum of 4y and -5xz 2 Binomial b.) 6.5 c.) 7a b d.) 6x 3 + 4x + x + 3 Standard form of a polynomial: the terms are in order from degree. Leading coefficient: in form, the coefficient of the term. Example 2: Write each polynomial in standard form. Identify the leading coefficient. a) 9x 2 + 3x 6 4x b) 34x + 9x 4 + 3x 7 4x 2 2

3 Example 3: Add the polynomials. a) (7y 2 + 2y 3) + (2 4y + 5y 2 ) b) (4x 2 2x + 7) + (3x 7x 2 9) Example 4: Subtract the polynomials. a) (6y 2 + 8y 4 5y) (9y 4 7y + 2y 2 ) b) (6n n 3 + 2n) (4n 3 + 5n 2 ) Example 5: The profit a business makes is found by subtracting the cost to produce an item C from the amount earned in sales S. The cost to produce and the sales amount could be modeled by the following equations, where x is the number of items produced. C = 100x x 300 S = 150x x a) Find an equation that models the profit. b) Use the above equation to predict the profit if 30 items are produced and sold. 3

4 Section 8.1 Worksheet Find each sum or difference. 1. (4y + 5) + ( 7y 1) 2. ( x 2 + 3x) (5x + 2x 2 ) 3. (4k 2 + 8k + 2) (2k + 3) 4. (2m 2 + 6m) + (m 2 5m + 7) 5. (5a 2 + 6a + 2) (7a 2 7a + 5) 6. ( 4p 2 p + 9) + (p 2 + 3p 1) 7. (x 3 3x + 1) (x x) 8. (6x 2 x + 1) ( 4 + 2x 2 + 8x) 9. (4y 2 + 2y 8) (7y y) 10. (w 2 4w 1) + ( 5 + 5w 2 3w) Determine whether each expression is a polynomial. If it is a polynomial, find the degree and determine whether it is a monomial, binomial, or trinomial a 2 b + 3b 2 a 2 b y3 + y g 2 h 3 k 14. x + 3x4 21x 2 x 3 Write each polynomial in standard form. Identify the leading coefficient x x x 7 + x 4 + 4x x x 3 x 18. 4x + 2x 5 6x BUSINESS The polynomial s 3 70s s 10,800 models the profit a company makes on selling an item at a price s. A second item sold at the same price brings in a profit of s 3 30s s Write a polynomial that expresses the total profit from the sale of both items. 20. GEOMETRY The measures of two sides of a triangle are given. If P is the perimeter, and P = 10x + 5y, find the measure of the third side. 4

5 Section 8.2 Notes: Multiplying a Polynomial by a Monomial Warm-Up Your fabulous math teacher had a really rough day and made a few errors. Please find them, correct the problems, and write out the error(s) in a complete sentence. 1) Simplify. 2) Simplify. 19 x 2 5 x 2 2 x 7 x x 2 8x 9 4x 2 3x + 1 ( + + ) + ( + ) ( ) ( ) = 26x 4 + 7x 2 2 = 9x 2 11x - 8 Example 1: Find each product. a) 6y(4y 2 9y 7) b) 3x(2x 2 + 3x + 5) Example 2: Simplify each expression. a) 3(2t 2 4t 15) + 6t(5t + 2) b) 5(4y 2 + 5y 2) + 2y(4y + 3) Example 3: Admission to the Super Fun Amusement Park is \$10. Once in the park, super rides are an additional \$3 each and regular rides are an additional \$2. Wyome goes to the park and rides 15 rides, of which s of those 15 are super rides. a) Write an equation to represent the total cost Wyome spent at the amusement park. b) Find the cost if Wyome rode 9 super rides. Example 4: Solve each equation. a) b(12 + b) 7 = 2b + b( 4 + b) b) d(d + 3) d(d 4) = 9d 16 5

6 Section 8.2 Worksheet Find each product. 1. 2h( 7h 2 4h) 2. 6pq(3p 2 + 4q) 3. 5jk(3jk + 2k) 4. 3rt( 2t 2 + 3r) m(8m2 + m 7) n2 ( 9n 2 + 3n + 6) Simplify each expression. 7. 2l(3l 4) + 7l 8. 5w( 7w + 3) + 2w( 2w w + 2) 9. 6t(2t 3) 5(2t 2 + 9t 3) 10. 2(3m 3 + 5m + 6) + 3m(2m 2 + 3m + 1) 11. 3g(7g 2) + 3(g 3 + 2g + 1) 3g( 5g + 3) Solve each equation (2t 1) + 3 = 3(3t + 2) 13. 3(3u + 2) + 5 = 2(2u 2) 14. 4(8n + 3) 5 = 2(6n + 8) (3b + 1) = 4(b + 3) t(t + 4) 1 = t(t + 2) u(u 5) + 8u = u(u + 2) 4 6

7 18. NUMBER THEORY Let x be an integer. What is the product of twice the integer added to three times the next consecutive integer? 19. INVESTMENTS Kent invested \$5000 in a retirement plan. He allocated x dollars of the money to a bond account that earns 4% interest per year and the rest to a traditional account that earns 5% interest per year. a. Write an expression that represents the amount of money invested in the traditional account. b. Write a polynomial model in simplest form for the total amount of money T Kent has invested after one year. (Hint: Each account has A + IA dollars, where A is the original amount in the account and I is its interest rate.) c. If Kent put \$500 in the bond account, how much money does he have in his retirement plan after one year? 20. COLLEGE Troy s boss gave him \$700 to start his college savings account. Troy s boss also gives him \$40 each month to add to the account. Troy s mother gives him \$50 each month, but has been doing so for 4 fewer months than Troy s boss. Write a simplified expression for the amount of money Troy has received from his boss and mother after m months. 21. MARKET Sophia went to the farmers market to purchase some vegetables. She bought peppers and potatoes. The peppers were \$0.39 each and the potatoes were \$0.29 each. She spent \$3.88 on vegetables, and bought 4 more potatoes than peppers. If x = the number of peppers, write and solve an equation to find out how many of each vegetable Sophia bought. 22. GEOMETRY Some monuments are constructed as rectangular pyramids. The volume of a pyramid can be found by multiplying the area of its base B by one third of its height. The area of the rectangular base of a monument in a local park is given by the polynomial equation B = x 2 4x 12. a. Write a polynomial equation to represent V, the volume of a rectangular pyramid if its height is 10 centimeters. b. Find the volume of the pyramid if x = 12. 7

8 Section 8.3 Day 1 Notes: Multiplying Polynomials Warm-Up 1) Simplify: 3w(w 2 + 7w 9) 2) Solve: 3(2x 3) 1 = 4(2x + 1) + 8 Double Distribute: a method used for multiplying. Quadratic expression: an expression in one variable with a expressions.. This happens when multiplying two linear Example 1: Find each product using the Double Distribute method. a) (z 6)(z 12) b) (5x 4)(2x + 8) c) (3x + 5)(2x 6) Example 2: Find the product using the box/area method. 1) ( x 2)( x+ 4) 2) ( x 3)( x+ 3) 3) (2x+ 1)( x+ 5) 8

9 Example 3: A patio in the shape of the triangle shown is being built in Lavali s backyard. The dimensions given are in feet. The area A of the triangle is one half the height h times the base b. Write an expression for the area of the patio. Example 4: The area of a rectangle is the measure of the base times the height. Write an expression for the area of the rectangle. 9

10 Section 8.3 Day 2 Notes: Multiplying Polynomials Warm-Up Find the product of the following binomials. 1) (2x 4)(3x + 2) 2) (-4x 8)(5x + 2) Example 5: Find the product of (3a + 4)(a 2 12a + 1) using the Distributive Property and the box/area method. Distributive Property Box/Area Method a 2 12a 1 3a 4 Example 6: Find the product using your method of choice. a) (2b 2 + 7b + 9)(b 2 + 3b 1) b) (3z + 2)(4z 2 + 3z + 5) 10

11 Section 8.3 Worksheet Find each product. 1. (q + 6)(q + 5) 2. (x + 7)(x + 4) 3. (4b + 6)(b 4) 4. (2x 9)(2x + 4) 5. (6a 3)(7a 4) 6. (2x 2)(5x 4) 7. (m + 5)(m 2 + 4m 8) 8. (t + 3)(t 2 + 4t + 7) 9. (2h + 3)(2h 2 + 3h + 4) 10. (3d + 3)(2d 2 + 5d 2) 11. (3q + 2)(9q 2 12q + 4) 12. (3r + 2)(9r 2 + 6r + 4) 13. (3n 2 + 2n 1)(2n 2 + n + 9) 14. (2t 2 + t + 3)(4t 2 + 2t 2) 11

12 GEOMETRY Write an expression to represent the area of each figure NUMBER THEORY Let x be an even integer. What is the product of the next two consecutive even integers? 18. GEOMETRY The volume of a rectangular pyramid is one third the product of the area of its base and its height. Find an expression for the volume of a rectangular pyramid whose base has an area of 3x x + 9 square feet and whose height is x + 3 feet. 19. THEATER The Loft Theater has a center seating section with 3c + 8 rows and 4c 1 seats in each row. Write an expression for the total number of seats in the center section. 20. ART The museum where Julia works plans to have a large wall mural replica of Vincent van Gogh s The Starry Night painted in its lobby. First, Julia wants to paint a large frame around where the mural will be. The mural s length will be 5 feet longer than its width, and the frame will be 2 feet wide on all sides. Julia has only enough paint to cover 100 square feet of wall surface. How large can the mural be? a. Write an expression for the area of the mural. b. Write an expression for the area of the frame. c. Write and solve an equation to find how large the mural can be. 12

13 Section 8.4 Notes: Special Products Warm-Up Find the product of each of the following. 1) (a + 6)(a 3) 2) (5b 3)(5b 2 + 3b 2) 3) Find an expression that represents the area of the figure below. Square of a Sum: Example 1: Square of a sum Find each product. a) (7z + 2) 2 b) (3x + 2) 2 Square of a Difference: Example 2: Square of a difference Find each product. a) (3c 4) 2 b) (2m 3) 2 13

14 Example 3: Write an expression that represents the area of a square that has a side length of 3x + 12 units. Product of a Sum and a Difference: Example 4: Product of a sum and difference Find each product. a) (9d + 4)(9d 4) b) (3y + 2)(3y 2) 14

15 Section 8.4 Worksheet Find each product. 1. (n + 9) 2 2. (q + 8) 2 3. (x 10) 2 4. (r 11) 2 5. (b + 6)(b 6) 6. (z + 13)(z 13) 7. (4j + 2) 2 8. (5w 4) 2 9. (3g + 9h)(3g 9h) 10. (4q + 5t)(4q 5t) 11. (a + 6u) (5r + p) (6h m) (4b 7v) (6a 7b)(6a + 7b) 16. (8h + 3d)(8h 3d) 17. (9x + 2y 2 ) (2b 2 g)(2b 2 + g) 19. (3p 3 + 2m) (5b 2 2b) (4m 3 2t) 2 15

16 22. GEOMETRY Janelle wants to enlarge a square graph that she has made so that a side of the new graph will be 1 inch more than twice the original side g. What trinomial represents the area of the enlarged graph? 23. GRAVITY The height of a penny t seconds after being dropped down a well is given by the product of (10 4t) and (10 + 4t). Find the product and simplify. What type of special product does this represent? 24. BUSINESS The Combo Lock Company finds that its profit data from 2005 to the present can be modeled by the function y = 4n n + 121, where y is the profit n years since Which special product does this polynomial demonstrate? Explain. 25. STORAGE A cylindrical tank is placed along a wall. A cylindrical PVC pipe will be hidden in the corner behind the tank. See the side view diagram below. The radius of the tank is r inches and the radius of the PVC pipe is s inches. a. Use the Pythagorean Theorem to write an equation for the relationship between the two radii. Simplify your equation so that there is a zero on one side of the equals sign. b. Write a polynomial equation you could solve to find the radius s of the PVC pipe if the radius of the tank is 20 inches. 16

17 Review Write the polynomial in standard form. Then give the degree and leading coefficient x+ 5 4x + x 2. 9x 10 + x+ 6x Standard Form: Standard Form: Degree: Leading Coefficient: Degree: Leading Coefficient: Simplify each expression. Write your answer in standard form. 3. (d 2 d + 5) (2d + 5) 4. (6k 2 + 2k + 9) + (4k 2 5k) 5. (x 2 3x) (2x 2 + 5x) 6. 3n 2 ( 2n 2 + 3n + 4) 7. 4b( 5b 3) 2b 2 (b 2 7b 4) 8. (3b + 3)(3b 2) 9. (5a 2)(2a 3) 10. (m + 3)( m 2 + 3m + 5) 11. (3t 1)(t 2 + 2t + 4) 12. (3g + 2)(3g 2) 13. (n + 3) (2k 2) 2 17

18 15. Solve: w(4w + 6) + 2w = 2(2w 2 + 7w 3) 16. Write an expression for the area of the triangle: 17. A parking garage charges \$0.50 per daytime hour and \$0.25 per hour during nights and weekends. Trent paid a total of \$5.00 to park throughout the week. He also spent 2 more hours in the parking garage during the night than during the daytime. If h = the number of daytime hours in the parking garage, write and solve an equation to find out how many hours Trent parked during the day and during the night/weekend. 18

19 Section 8.5 Day 1 Notes: Using the Distributive Property (GCF) Warm-Up List the factors of each of the numbers. Then circle the greatest (biggest) one they have in common. 1) 15, 20, 45 2) 36, 18, 6 3) 21, 49, 14 4) 8, 24, 4 Factoring: to express a polynomial as the product of monomials and polynomials. GCF Basics To find the GCF of a number: 1. Ask yourself if the number will divide into the other numbers. a. If it does, that s your GCF! b. If it doesn t, keep looking. Think of (or list) factors of the number. Of those factors, find the number that will divide into all the numbers. Example 1: Find the GCF of the numbers. a) 6, 18, 14 b) 24, 3, 12 c) 8, 24, 28 To find the GCF of a variable: 1. Take the number of variables possible. 2. If even one term does a variable, don t include any variables in your GCF. Example 2: Find the GCF of the variables. a) x 6 + x 2 + x 7 b) 1 + x 4 + x 9 c) x + x 4 + x 3 - x 2 Things to remember: Make sure your polynomial is in first. If your leading coefficient is, make your negative. Example 1: Use the Distributive Property to factor each polynomial. a) 15x + 25x 2 b) 4x 2 y 2 + 2x 3 y 3 + 8xy 4 c) 9p 3 + 6p

20 Additional Examples: Factor the greatest common factor out of the following polynomials. 1) 4p p 2) 18d 6 6d 2 + 3d 3) 33x x 2 4) 5x 3 25x 2 35x 5) 4x ) 75x 4 y + 3x 2 7) -3w w 3 8) 24t 7 60t t 4 36t 9) 14t 4 35t 21 8dq + 4dq+ 12dq 11) 5c 25c + 10c 12) 15y + 6y 21y 10) ) 5 xy + 25x y 14) xz 24xz 15) 4 10y 2y 2 16) a 28a 17) 2 32x 48x 18) 35xy 60x 2 19) 3m m ) 4x 4 + 4x 80 21) 2t + 2t 12t

21 Section 8.5 Day 2 Notes: Using the Distributive Property (Zero Product Property) Warm-Up Factor out the GCF in the following problems 1) 3x 6 2) 5x 2 10x 3) 12x 3 + 6x 2 4x Zero Product Property: Example 2: Solve each equation using the zero product property. Check each solution. a) (x 2)(4x 1) = 0 b) 12y 2 = 4y c) (s 3)(3s + 6) = 0 d) 40x 2 5x = 0 Example 3: A football is kicked into the air. The height of the football can be modeled by the equation h = 16x x, where h is the height reached by the ball after x seconds. Find the values of x when h = 0. Example 4: Juanita is jumping on a trampoline in her back yard. Juanita s jump can be modeled by the equation h = 14t t, where h is the height of the jump in feet at t seconds. Find the values of t when h = 0. 21

22 Section 8.5 Worksheet Factor the GCF out of the following polynomials ab 2. 4d r 2 t 3rt x c y 2 + 4y x 2x + 2x 8. d d 2d 9. 10y 12y + 4y 3 2 Solve each equation. Check your solutions. 10. x(x 32) = b(b + 4) = (y 3)(y + 2) = (a + 6)(3a 7) = z z = p 2 4p = x 2 = 27x x 2 = 15x 18. 8x 2 = 26x 19. LANDSCAPING A landscaping company has been commissioned to design a triangular flower bed for a mall entrance. The final dimensions of the flower bed have not been determined, but the company knows that the height will be two feet less than the base. The area of the flower bed can be represented by the equation A = 1 2 b2 b. a. Write this equation in factored form. b. Suppose the base of the flower bed is 16 feet. What will be its area? 22

23 20. PHYSICAL SCIENCE Mr. Alim s science class launched a toy rocket from ground level with an initial upward velocity of 60 feet per second. The height h of the rocket in feet above the ground after t seconds is modeled by the equation h = 60t 16t 2. How long was the rocket in the air before it returned to the ground? 21. PHYSICS According to legend, Galileo dropped objects of different weights from the so-called leaning tower of Pisa while developing his formula for free falling objects. The relationship that he discovered was that the distance d an object falls after t seconds is given by d = 16t 2 (ignoring air resistance). This relationship can be found in the equation h = 4t 16t 2, where h is the height of an object thrown upward from ground level at a rate of 32 feet per second. Solve the equation for h = SWIMMING POOL The area A of a rectangular swimming pool is given by the equation A = 12w w 2, where w is the width of one side. Write an expression for the other side of the pool. 23. CONSTRUCTION Unique Building Company is constructing a triangular roof truss for a building. The workers assemble the truss with the dimensions shown on the diagram below. Using the Pythagorean Theorem, find the length of the sides of the truss. 24. VERTICAL JUMP Your vertical jump height is measured by subtracting your standing reach height from the height of the highest point you can reach by jumping without taking a running start. Typically, NBA players have vertical jump heights of up to 34 inches. If an NBA player jumps this high, his height h in inches above his standing reach height after t seconds can be modeled by h = 162t 192t 2. Solve the equation for h = 0 and interpret the solution. Round your answer to the nearest hundredth. 23

24 Section 8.6 Day 1 Notes: Factoring x 2 + bx + c Warm-Up Solve for y in the following equations. 1) 16y 2 20y = 0 2) 6y 2 = 9y What is different about the two equations above? Why does this matter? You have learned how to multiply two binomials. Each of the binomials was a factor of the product. Example 1: Factor each polynomial. a) x 2 + 7x + 12 b) x 2 12x + 27 c) x 2 + 3x 18 24

25 Section 8.6 Day 2 Notes: Factoring x 2 + bx + c Warm-Up Factor each of the following expressions. 1) x 2 6x 16 2) x 2 + 2x 8 Example 2: Factor each polynomial. Remember to check for the GCF first!! a) 2x x + 16 b) 3x 2 6x 45 c) 2x x + 24 Extra Practice: Factor each polynomial. Remember to check for the GCF first!! a) x 2 10x + 16 b) x 2 x 20 c) x 2 + 5x + 6 d) x 2 7x 18 e) 5x x + 20 f) 4x 2 4x 8 g) x 2 7x + 10 h) x x + 12 i) 2x 2 + 2x 4 25

26 Section 8.6 Day 3 Notes: Solving x 2 + bx + c = 0 Warm-Up Factor each of the following expressions. 1) x 2 7x ) 2x 2 + 4x 30 Quadratic Equation: can be written in the standard form, where a 0. Example 3: Solve each equation. Check your solutions. a) x 2 + 2x = 15 b) x 2 20 = x c) x 2 + 4x + 27 = 24 Example 4: Solve each equation. Check your solutions. a) 4x 2 + 4x = 24 b) 2x 2 6x = 20 c) 2x = 22x Example 5: Marion wants to build a new art studio that has three times the area of her old studio by increasing the length and width by the same amount. What should be the dimensions of the new studio? Example 6: Adina has a 4 6 photograph. She wants to enlarge the photograph by increasing the length and width by the same amount. What dimensions of the enlarged photograph will produce an area twice the area of the original photograph? 26

27 Section 8.6 Worksheet Factor each polynomial. 1. a a h h x x g 2 2g w 2 + w b 2 + 4b n 2 6n t 2 + 4t z 2 44z q 2 q x 2 6x r + r 2 Solve each equation. Check the solutions. 13. x x + 42 = p 2 + 5p 84 = k 2 + 3k 54 = n 2 + 4n = r + 2r 2 = a 2 = 18a 19. 3t 2 78t = z 2 14z = y 2 84 = 5y 27

28 22. CONSTRUCTION A construction company is planning to pour concrete for a driveway. The length of the driveway is 16 feet longer than its width w. a. Write an expression for the area of the driveway. b. Find the dimensions of the driveway if it has an area of 260 square feet. 23. WEB DESIGN Janeel has a 10-inch by 12-inch photograph. She wants to scan the photograph, then reduce the result by the same amount in each dimension to post on her Web site. Janeel wants the area of the image to be one eighth that of the original photograph. a. Write an equation to represent the area of the reduced image. b. Find the dimensions of the reduced image. 24. COMPACT DISCS A compact disc jewel case has a width 2 centimeters greater than its length. The area for the front cover is 168 square centimeters. The first two steps to finding the value of x are shown below. Solve the equation and find the length of the case. Length width = area x(x + 2) = 168 x 2 + 2x 168 = 0 28

29 Section 8.7 Day 1 Notes: Factoring ax 2 + bx + c Warm-Up Solve each of the following quadratic equations. 1) x 2 + 8x = 12 2) 3x x + 9 = 0 In the last lesson, you factored expressions where a = 1. This lesson focuses on when a 1. Steps for Slide and Divide 1. Slide the a over to c and them together. 2. Rewrite in the form x 2 + bb + (aa). 3. Factor using. 4. Divide by a. 5. any fractions. 6. Slide any back in front of the x. We will use the Slide and Divide Method. Example 1: Factor each trinomial using the Slide and Divide Method. Remember to look for the GCF first!! a) 5x x + 10 b) 7x x + 4 c) 4x x + 4 d) 2x 2 x 1 29

30 Prime Polynomial: a polynomial that cannot be written as a product of two polynomials with coefficients that are. Example 2: Factor each trinomial. Determine whether the polynomial is prime. a) 3x 2 + 7x 5 b) 3x 2 5x + 3 Extra Practice: Factor each trinomial. Remember to look for the GCF first!! a) 2x 2 + 5x + 3 b) 4x x + 32 c) 2x x + 20 d) 4x 2 + 3x 1 e) 5x x + 1 f) 4x x

31 Section 8.7 Day 2 Notes: Solving ax 2 + bx + c = 0 Warm-Up Factor each of the following expressions. 1) 3x 2 + 5x + 2 2) 6x 2 4x 16 Example 3: Solve each equation. Check your solutions. a) 3x 2 4x = 15 b) 3x = 21x c) 3x x + 9 = 1 d) 8x x 11 = 3x 10 31

32 Rally Coach 1) Factor 2x 2 + 7x + 6 2) Solve 3x 2 + 5x + 2 = 0 3) Factor 2x 2 5x 3 4) Solve 3x x = 8 5) Factor 3x x + 3 6) Solve 2x 2 + 3x 5 = 0 7) Factor 3x 2 + 9x 12 8) Solve 2x 2-2x = 4 32

33 Section 8.7 Day 3 Notes: Solving ax 2 + bx + c = 0 (Vertical Motion) Warm-Up 1) Solve 10x 2 13x + 3 = 0 2) Solve 10x = 26x Example 4: When Mario jumps over a hurdle, his feet leave the ground traveling at an initial upward velocity of 12 feet per second. Find the time t in seconds it takes for Mario s feet to reach the ground again. Use the equation h = 16t 2 + vt + h 0, where h is height in feet, t is time in seconds, v is the initial upward velocity in feet per second, and h o is the initial height in feet. Example 5: Mr. Nguyen s science class built a model rocket for a competition. When they launched their rocket outside the classroom, the rocket cleared the top of a 60-foot high pole and then landed in a nearby tree. If the launch pad was 2 feet above the ground, the initial velocity of the rocket was 64 feet per second, and the rocket landed 30 feet above the ground, how long was the rocket in flight? Use the equation h = 16t 2 + vt + h 0, 33

34 Vertical Motion Extra Problems h = 16t 2 + vt + h o 1. A startled armadillo jumps straight into the air with an initial vertical velocity of 14 feet per second. After how many seconds does it land on the ground? 2. A cat leaps from the ground into the air with an initial vertical velocity of 11 feet per second. After how many seconds does the cat land on the ground? 3. An athlete throws a discus from an initial height of 6 feet and with an initial vertical velocity of 46 feet per second. After how many seconds does the discus hit the ground? 4. You throw a ball into the air with an initial vertical velocity of 31 feet per second. The ball leaves your hand when it is 6 feet above the ground. You catch the ball when it reaches a height of 4 feet. After how many seconds do you catch the ball? 5. You hit a baseball straight up into the air. The baseball is hit with an initial vertical velocity of 80 feet per second when it is 3 feet off the ground. After how many seconds does the ball reach a height of 99 feet? 34

35 Section 8.7 Worksheet Factor each polynomial, if possible. If the polynomial cannot be factored using integers, write prime. 1. 2b b g 2 + 8g x 2 + 4x b 2 5b m 2 + 7m d d y 2 4y k 2 9k z z q q h h p 2 22p 20 Solve each equation. Check the solutions h 2 + 2h 16 = n 2 n = q 2 10q + 3 = b 2 5b = r 2 21r = 4r g = 29g x 2 1 = x 20. 8a 2 16a = 6a a a = 11a

36 22. DIVING Lauren dove into a swimming pool from a 15-foot-high diving board with an initial upward velocity of 8 feet per second. Find the time t in seconds it took Lauren to enter the water. Use the model for vertical motion given by the equation h = 16t 2 + vt + h o, where h is height in feet, t is time in seconds, v is the initial upward velocity in feet per second, and h o is the initial height in feet. (Hint: Let h = 0 represent the surface of the pool.) 23. BASEBALL Brad tossed a baseball in the air from a height of 6 feet with an initial upward velocity of 14 feet per second. Enrique caught the ball on its way down at a point 4 feet above the ground. How long was the ball in the air before Enrique caught it? Use the model of vertical motion from Exercise BREAK EVEN Breaking even occurs when the revenues for a business equal the cost. A local children s museum studied their costs and revenues from paid admission. They found that their break-even time is given by the equation 2h 2 2h 24 = 0, where h is the number of hours the museum is open per day. How many hours must the museum be open per day to reach the break even point? 25. FURNITURE The student council wants to purchase a table for the school lobby. The table comes in a variety of dimensions, but for every table, the length is 1 meter greater than twice the width. The student council has budgeted for a table top with an area of exactly 3 square meters. Find the width and length of the table they can purchase. 36

37 Factoring/Solving All Types Homework ( ) Factor the expression or solve the equation. Show all work. 1.) x 2 5x 14 2.) 4x 3 12x 2 3.) 2x 2 + 3x 9 4.) x 2 + 5x 24 = 0 5.) 3x 2 8x + 4 = 0 6.) 4x x + 6 = 0 7.) 6x x 2 + 4x 8.) 2x 3 + 2x 2 12x 37

38 Review Determine the GCF for each of the following polynomials. Then, factor out the GCF. 1. 6x 2 + 3x 2. 8x Factor the polynomial completely. If the polynomial cannot be factored using integers, write prime. 3. 6x 2 + 5x 6 4. x 2 + 2x x 2 4x x 2 27x x x x 8. 2x x + 5 Solve the equation. 9. x(x 3) = x 2 + x = 2 38

39 11. 3x 2 8x 16 = x = 15x 13. 3x 2 = 9x 14. The length of a rectangle is 3 centimeters more than the width. The area of the rectangle is 40 square centimeters. What is the length? x x A pumpkin is launched into the air from ground level with an initial vertical velocity of 80 feet per second. How long does it take to hit the ground again? Use the equation h = 16t 2 + vt + h 0 39

40 Section 8.8 Notes: Differences of Squares Warm-Up Factor or solve the following: 1) 6x 2 + 5x 6 2) 20x 2 13x = 2 Difference of Two Squares: two perfect squares separated by a sign. Example 1: Factor each polynomial. a) m 2 64 b) 16y 2 81z 2 c) 25a 2 36b 2 Example 2: Factor each polynomial. a) 3b 3 27b b) 5x 3 20x Example 3: Factor each polynomial. a) y b) y 4 16 Example 4: Solve the equation. a) x 2 81 = 0 b) x 2 64 = 0 c) 9x 2 25 = 0 40

41 Section 8.8 Worksheet Factor each polynomial, if possible. If the polynomial cannot be factored, write prime. 1. k r p x f g 2 49h m 2 144p y a 2 54b t 2 18u d z 3 9z q 3 20q b 3 36b 15. 3t 4 48t 2 Solve each equation by factoring. Check your solutions y 2 81 = p 2 9 = b 2 50 = LOTTERY A state lottery commission analyzes the ticket purchasing patterns of its citizens. The following expression is developed to help officials calculate the likely number of people who will buy tickets for a certain size jackpot. 81a 2 36b 2 Factor the expression completely. 20. OPTICS A reflector on the inside of a certain flashlight is a parabola given by the equation y = x Find the points where the reflector meets the lens by finding the values of x when y = 0. 41

42 21. BALLOONING The function f (t) = 16t represents the height of a freely falling ballast bag that starts from rest on a balloon 576 feet above the ground. After how many seconds t does the ballast bag hit the ground? 22. DECORATING Marvin wants to purchase a rectangular rug. It has an area of 80 square feet. He cannot remember the length and width, but he remembers that the length was 8 more than some number and the width was 8 less than that same number. a. Write a quadratic equation using the information given. b. What are the length and width of the rug? 42

43 Section 8.9 Day 1 Notes: Perfect Squares Warm-Up Factor or solve the following: 1) x ) 25y ) 81x 2 9 = 0 Perfect Square Trinomial: a trinomial that is the square of a binomial. Example 1: Determine whether each trinomial is a perfect square trinomial. Write yes or no. If so, factor it. a) 25x 2 30x + 9 b) 49y y + 36 c) 9x 2 12x + 16 d) 49x x

44 A polynomial is completely factored when it is written as a product of prime polynomials. More than one method might be needed to factor a polynomial completely. Remember, if the polynomial does not fit any pattern or cannot be factored, the polynomial is prime. Example 2: Factor completely. a) 6x 2 96 b) 4x x + 6 c) 4x x + 9 Example 3: Solve each equation. Check your solution. a) 4x x = 81 b) 9x 2 30x + 25 = 0 44

45 Section 8.9 Day 2 Notes: Perfect Squares Warm-Up Factor each of the following: 1) x 2 8x 9 2) 25x 2 9 3) 16x 2 40x + 25 Solve for x. 1. x 2 = x 2 = x = x 2 = 28 Square Root Property: Example 4: Solve each equation using the square root property. Check your solutions. a) (b 7) 2 = 36 b) (x + 9) 2 = 8 c) (x 4) 2 = 25 Example 5: A book falls from a shelf that is 5 feet above the floor. A model for the height h in feet of an object dropped from an initial height of h 0 feet is h = 16t 2 + h 0, where t is the time in seconds after the object is dropped. Use this model to determine approximately how long it took for the book to reach the ground. 45

46 Section 8.9 Worksheet Determine whether each trinomial is a perfect square trinomial. Write yes or no. If so, factor it. 1. m m r 2 6r y 2 20y p p b 2 4b k 2 56k + 16 Factor each polynomial, if possible. If the polynomial cannot be factored, write prime. 7. 3p x x q 2 60q t 3 14t 2 12t 11. 6d k k b 2 24bf h 2 60h n 2 30n u 2 28m w 4 8w a ad + 81d 2 Solve each equation. Check the solutions k 2 28k = b b + 2 = x x + 36 = y 2 8y + 16 = (h + 9) 2 = w 2 6w + 9 = 13 46

47 25. PICTURE FRAMING Mikaela placed a frame around a print that measures 10 inches by 10 inches. The area of just the frame itself is 69 square inches. What is the width of the frame? 26. CONSTRUCTION The area of Liberty Township s square playground is represented by the trinomial x 2 10x Write an expression using the variable x that represents the perimeter. 27. AMUSEMENT PARKS Funtown Downtown wants to build a vertical motion ride where the passengers are launched straight upward from ground level with an initial velocity of 96 feet per second. The ride car s height h in feet after t seconds is h = 96t 16t 2. How many seconds after launch would the car reach 144 feet? 28. BUSINESS Saini Sprinkler Company installs irrigation systems. To track monthly costs C and revenues R, they use the following functions, where x is the number of systems they install. R(x) = 8x x + 4 C(x) = 7x x 12 The monthly profit can be found by subtracting cost from revenue. P(x) = R(x) C(x) Find a function to project monthly profit and use it to find the break-even point where the profit is zero. 29. GEOMETRY Holly can make an open-topped box out of a square piece of cardboard by cutting 3-inch squares from the corners and folding up the sides to meet. The volume of the resulting box is V = 3x 2 36x + 108, where x is the original length and width of the cardboard. b. What is the volume of the box if the original length of each side of the cardboard was 14 inches? a. Factor the polynomial expression from the volume equation. c. What is the original side length of the cardboard when the volume of the box is 27 in 3? 47

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