Chapter 4 Factoring and Quadratic Equations Lesson 1: Factoring by GCF, DOTS, and Case I Lesson : Factoring by Grouping & Case II Lesson 3: Factoring by Sum and Difference of Perfect Cubes Lesson 4: Solving Quadratic Equations by Factoring Lesson 5: Solving Quadratic Inequalities This assignment is a teacher-modified version of Algebra Common Core Copyright (c) 016 emath Instruction, LLC used by permission.
Chapter 4 Lesson 1 GCF, DOTS, and Case I Factoring Factoring: When we factor it is important to remember that ; we are simply rewrite it in an equivalent form. GCF or Greatest Common Factor : Factoring by GCF means that we what terms have in. This can be a combination of numbers, variables, or both. Exercise #1: Factor out by GCF. (a) 3x + 6x (b) 0x 5x (c) 10x y 3 5xy 4 (d) 15x 30x 3 (e) x + 8x + 10 (f) 8x 3 1x + 0x
Exercise #: Rewritten in factored form 0x 36x is equivalent to (1) x(10x 15) (3) 5x(4x + 7) () 4x(5x 9) (4) 9x(x 4) Conjugate Multiplication Pattern DOTS Factoring: Another type of factoring stems from multiplication. This type of factoring is known as Difference of Two Squares or DOTS factoring. When we factor these types of expressions, we conjugate multiplication. Exercise #3: Write each of the following binomials as the product of a conjugate pair. (a) x 9 (b) 5 4x (c) (d)
Exercise #4: Factor each expression completely. Factoring Completely: When we factor completely, it means that we factor until we. It is important to always look for a GCF first. (a) 8x 7 (b) 3x 3 48x (c) 16x 4 81 (d) x 4 16 Trinomial Case I Factoring: Another type of factoring is trinomial factoring. This is when we have a trinomial. In Case I factoring, the leading coefficient is. To factor these, it is helpful to look at the term and the term. Exercise #5: Write each of the following trinomials in factored form. (a)x 7x 18 (b) x + 14x + 4
(c) x + x 1 (d) x 5x + 6 (e) x 15x + 44 (f) x 6x 16 Exercise #6: Factor each expression completely. (a) 4x + 8x + 4 (b) 5x - 5x 30 (c) x + 8x 64 (d) 3x + 18x + 7
Chapter 4 Lesson 1 HOMEWORK GCF, DOTS, and Case I Factoring Fluency: 1. Factor the following completely: (a) 30x 35x (b) 3 0 5 15 3 x x x (c) 14x 35x 7x (d) 3 7x 1 x (e) 8x 51 (f) 5x 9 1 (h) (i) (k) x 17x 30
. If one factor of 56x 4 y 3 4x y 6 is 14x y 3, what is the other factor? (1) 4x 3y 3 (3) 4x y 3xy 3 () 4x 3y (4) 4x y 3xy 3. When factored completely, x 3 13x 30x is (1) x(x + 3)(x 10) (3) x(x + )(x 15) () x(x 3)(x 10) (4) x(x )(x + 15) Applications 4. The area of any rectangular shape is given by the product of its width and length. If the area of a particular rectangular garden is given by A 15x 35x expression for the garden s length. Justify your response. and its width is given by 5x, then find an 3 5. The volume of a particular rectangular box is given by the equation V 50x x. The height and length of the box are shown on the diagram below. Find the width of the box in terms of x. Recall that V L W H for a rectangular box. x x 5?
Chapter 4 Lesson Grouping & Case II Factoring Factoring by Grouping: A new type of factoring is factoring by grouping. This type of factoring requires us to see structure in expressions. We usually factor by grouping when we have a polynomial that has four or more terms. Example x 3 + x + 3x + 6 Steps 1. terms together that have a factor.. Factor each group 3. Factor by 4. Distribute to check (if needed) Exercise #1: Factor the expression x 3 6x + 5x 15. Justify each step with one of the three major properties of real numbers, i.e. the commutative, associative, or distributive.
Exercise #: Use the method of factoring by grouping to completely factor the following expressions. (a) 3x 3 + x - 7x - 18 (b) 18x 3 + 9x - x - 1 (c) x 5 + 4x 3 + x + 8 (d) 5x 3 + 10x + 0x + 40 Exercise #3: Write the expression (x + 3)(x - 4) + 5(x + 3) as the equivalent product of binomials. Exercise #4: Consider the expression x + ab - ax - bx. (a) How can you rewrite the expression so that the first two terms share a common factor (other than 1)? (b) Write this expression as an equivalent product of binomials.
Exercise #5: Louis factored the expression x 3 + 10x + 7x + 1 below. Is he correct? Explain. x 3 + 10x + 7x + 1 = x (x+5) +7(x+3) = (x + 7)(x + 5 + x + 3) = (x + 7)(x + 8) Case II Trinomial Factoring: Case II Trinomial factoring is used when the leading coefficient is not 1, even after any GCF was taken out. Example x 7x + 6 Steps 1. See if you can factor out a GCF.. Multiply the coefficient of the first tem with the last term. 3. Split the middle term. You must determine the signs and coefficients for the two terms. 4.Factor by grouping. Exercise #6: Factor the following trinomials. (a) 3x + 19x - 40 (b) x - 15x + 18
(c) 15x + 13x + (d) 10x + 13x 30 (e) 1x + 8x - 15 (f) 36x - 35x + 6 Exercise #7: Factor the following trinomials completely. (a) 10x + 55x - 105 (b) 1x + 57x - 15 (c) x + 0x + 50 (d) 1x + 9x - 8
Chapter 4 Lesson HOMEWORK Grouping & Case II Factoring Fluency: 1. Rewrite each of the following as a product of binomials. (a) x 10 x 3 x 5 x 3 (b) 3x 7 x 5 x 5 x 4 (c) 10x 6x 35x 1 (d) 1x 3x 0x 5 (e) x 9x 15 (f) 18x 5x 8. Which of the following is the correct factorization of the trinomial 1x 3x 10? (1) 6x 1 3x 10 (3) 4x 5 3x () 6x x 5 (4) 4x 5 3x
3. Factor each expression completely. 3 (a) 15x 110x 10 (b) 10x 6x 1x Reasoning: 4. Consider the expression: 3 x x x 5 9 45. Enter this expression on your calculator and find its zeroes. Provide evidence. Then, factor it completely. Do you see the relationship between the factors and the zeroes?
Chapter 4 Lesson 3 Factoring Sum & Difference of Cubes Two special factoring formulas are the sum and difference of perfect cubes. These formulas are: a 3 + b 3 = a 3 b 3 = Yes, these formulas have to be memorized!!! The quadratic portion usually cannot be factored. Examples: When you have a pair of cubes, carefully apply the appropriate rule. By carefully, I mean using parentheses to keep track of everything, especially the negative signs. Here are some typical problems: Factor: 1.) x 3 8 Hint: It helps to write the example in cubed form. For this example, we would write this as: (x) 3 () 3. By doing this, we can clearly see which is our a value and which is our b value. Now, finish factoring by using the rules.
Exercise #: Factor each of the following. (a) a 3 + 7 (b) x 3 64 (c) 8b 3 + 16c 6 (d) x 9 15
Excerise #3: Factor each of the following completely. (a) x 3 + 18 (b) x 9 51 (c) x 6 79 (d) 686y 3 Exercise #4: Daniel incorrectly factored x 6 15 as (x 3 + 5)(x 3 5). Where did Daniel go wrong? What is the correct factorization?
Exercise #5: y 6 64 can be first factored as a difference of cubes or as a difference of squares. (a) Factor y 6 64 first as a difference of cubes, and then factor completely. (b) Factor y 6-64 first as a difference of squares and then factor completely. (c) Based on your answers to part (a) and part (b), if a polynomial can be factored as a difference of cubes or a difference of squares which one should you do first? Why?
Chapter 4 Lesson 3 Homework Factoring Sum & Difference of Cubes Fluency Exercise #1: Each of the following expressions is either a difference of perfect cubes or a sum of perfect cubes. Factor appropriately. (a) 16z 3 w 3 (b) 7r 3 + 1000s 3 (c) 50t 3 + 16s 3 (d) 8k 6 7q 3 (e) a 6 8b 3 (f) 64y 6 1
Exercise #: Louis incorrectly factored x 3 16 as (x + 6)(x 36). Where did Louis go wrong? What is the correct factorization? Exercise #3: Factor 16s 3 + 7t 3. (1) (6s + 3t)(36s 18st + 9t ) (3) (6s 3t)(36s + 18st + 9t ) () (9s + 3t)(36s + 9t ) (4) (6s + 3t)(36s + 18st + 9t )
Chapter 4 Lesson 4 Solving Quadratic Equations Solving Quadratic Equations The Zero Product Law: If the product of multiple factors is equal to zero then at least one of the factors must be equal to zero. The Zero Product Law can be used to solve any quadratic equation that is (not prime). To utilize this technique, we must first set the equation to and then factor the non-zero side. Exercise #1: Solve each of the following quadratic equations using the Zero Product Law. (a) x + 3x 14 = -x + 10 (b) 3x + 1x 7 = x + 3x
Exercise #: Consider the system of equations shown below consisting of a parabola and a line. y = 3x - 8x + 5 and y = 4x + 5 (a) Find the intersection points of these curves algebraically. (b) Using your calculator, sketch the graph of this system on the axes on the axes below. Be sure to label the curves with equations, the intersection points, and the window. (c)verify your answer to part (a) by using the Intersect command on your calculator.
Exercise #3: The parabola shown below has the equation y = x x 3. (a) Write the coordinates of the two x-intercepts of the graph. (b) Algebraically find the x-intercepts of the parabola. Exercise #4: Algebraically find the set of x-intercepts (zeros) for each parabola given below. (a) y = 4x 1 (b) y = 3x + 13x 10
(c) y = 4x 10x (c) y = x + 13x 14 Exercise #5: A quadratic function of the form y = x + bx + c is shown below. (a) What are the x-intercepts of this parabola? (b) Based on your answer in part (a), write the equation of this quadratic function first in factored form and then in trinomial form.
Chapter 4 Lesson 4 Homework Solving Quadratic Equations Fluency: 1. Solve each of the following equations for the value of x. (a) 1x 8x 0 (b) x x x 4 40 10 15 (c) 6x 15x x 10x 4 (d) 4x 3x 11 3x
Applications:. Consider the system of equations shown below consisting of one linear and one quadratic equation. y x y x x 4 5 and 5 10 (a) Find the intersection points of this system algebraically. y x (b) Using your calculator, sketch a graph of this system to the right. Be sure to label the curves with equations, the intersection points, and the window. (c) Verify with the Intersect function on your calculator.
3. Algebraically, find the zeroes (x-intercepts) of each quadratic function given below. (a) y 1x 18x (b) y x x 6 8 Reasoning 4. A quadratic function of the form y x bx c is shown on the graph below. y (a) What are the x-intercepts of this parabola? x (b) Based on your answer to part (a), write the equation of this quadratic function first in factored form and then in trinomial form.
Chapter 4 Lesson 5 Solving Quadratic Inequalities Quadratic Inequalities in One Variable: Quadratic inequalities are similar to quadratic equations. When solving these types of problems, it is important to first find the zeros of the quadratic equation. Then, we put these zeros on a number line and test each interval to see where we must shade. Finally, we write our answer as an inequality. < or >: we need to put an open circle on our number line. < or >: we need to put a close circle on our number line. Exercise #1: Solve each of the following quadratic inequalities. Graph your solutions on a number line and write your final answer in set-builder notation. (a) x 5x 36 < 0 (b) x x 1 > 0 (c) x 13x 7 > 0 (d) 5x + 8x 1 < 0
(e) x 4x 8 > 10x 8 (f) x + 14x 6 < 14x + 19 Exercise #: The number line graph inequalities? is the solution to which of the following (1) x x 8 > 0 (3) x x 8 > 0 () x + x 8 < 0 (4) x + x 8 < 0 Exercise #3: Which of the following represents the solution set of the inequality -x + 7x 3 > 0? (1) (3) () (4)
Chapter 4 Lesson 5 Homework Solving Quadratic Inequalities Fluency 1. Which of the following values of x is in the solution set of the inequality x x 0? (1) 1 (3) 0 () (4) 4. The solution set of the inequality x 5 is which of the following? (1) 5, (3), 5 5, () 5, 5 (4),5 3. The solution to the inequality x 9 0 can be expressed graphically as (1) (3) -5 0 5-5 0 5 () (4) -5 0 5-5 0 5
4. Find the solution set to each of the quadratic inequalities shown below. Represent your solution set using any acceptable notation and graphically on a number line. (a) x 5x 6 (b) x 10x 4 (c) 8x 50x 5 10x 5 (d) 7x 4x 3 3x 4x 4