Unit 8 Notes: Solving Quadratics by Factoring Alg 1 Name Period Day Date Assignment (Due the next class meeting) Tuesday Wednesday Thursday Friday Monday Tuesday Wednesday Thursday Friday Monday Tuesday Wednesday Thursday Friday Monday Tuesday Wednesday Thursday 3/4/14 (A) 3/5/14 (B) 3/6/14 (A) 3/7/14 (B) 3/10/14 (A) 3/11/14 (B) 3/12/14 (A) 3/13/14 (B) 3/14/14 (A) 3/17/14 (B) 3/18/14 (A) 3/19/14 (B) 3/20/14 (A) 3/21/14 (B) 3/24/14 (A) 3/25/14 (B) 3/26/14 (A) 3/27/14 (B) 8.1 Worksheet Intro to Polynomials 8.2 Worksheet Multiplying Binomials 8.3 Worksheet Factoring and Solving by GCF and Grouping 8.4 Worksheet Factoring and Solving Trinomials 8.5 Worksheet Practice Factoring Day 8.6 Worksheet Factoring and Solving Trinomials with a leading coefficient 8.7 Worksheet More Factoring and Solving Unit 8 Practice Test Unit 8 Test NOTE: You should be prepared for daily quizzes. Every student is expected to do every assignment for the entire unit, or else Homework Club will be assigned! (Students with 100% Homework completion at the end of the semester will be rewarded with a pizza party and 2% grade increase.) HW reminders: If you cannot solve a problem, get help before the assignment is due. Need help? Try DRHSalgebra1.weebly.com. Extra Help? Visit www.mathguy.us or www.khanacademy.com. 8.1 Introduction to Polynomials Monomial Binomial Trinomial Polynomial Note: All the exponents must be numbers! 1
Degree of a polynomial Leading Coefficient Descending order 2 nd Differences. Tell whether each is a linear or quadratic function x 1 2 3 4 x 1 2 3 4 y 9 12 17 24 y 13 9 5 1 Adding polynomials Example 1: (4x 3 + x 2 5) + (7x + x 3-3x 2 ). Example 2: Find the sum: (x 2 + x + 8) + (x 2 x l) Subtracting polynomials Example 3: Find the difference: (4z 2-3) (-2z 2 + 5z - l). Remember to multiply each term in the polynomial by 1 when you write the subtraction as addition. Example 4: Find the difference of (3x 2 + 6x 4) (x 2 x 7). 2
Objective #1: Can you add and subtract polynomials? a) (3m 3 + 2m + 1) + (4m 2 3m + 1) b) ( 4c + c 3 + 8) + (c 2 5c 3) c) (3x 2 + 5) (x 2 + 2) + ( 3m + 1) d) ( 3z + 6) (4z 2 7z 8) 2 good ways to multiply polynomials: 1. Use distribution properties 2. Use box method Step 1: Decide the dimensions of the box your will draw. by # of terms Step 2: Draw the box and label each. Step 3: Multiply the terms for each part of your box. Step 4: Combine like terms and write your answer in descending order. Example 5: Find the product 3x 3 (2x 3 x 2 7x 3). Example 6: Multiply x 2 (x 6) 3
Objective #2: Can you multiply monomials and polynomials? a) 4x(x 3) b) 4x 2 (3x 4 2x + 7) c) 9x(x 2 + 5) 8.2: Multiplying Polynomials Warm-Up Simplify the following: 1) 2x 3 3x 5 2x 2) 3(2x 9) 3) (2s 3 s 2 + 1) (3s 2 s + 4) Multiplying polynomials (Box Method) Step 1: Decide the dimensions of the box your will draw. by Step 2: Draw the box and label each. Step 3: Multiply the terms for each part of your box. Step 4: Combine like terms and write your answer in descending order. Example 1: a. (2a 5)(a 2 6a 3) b. (3b + 5)(5b 6) 4
Alternative to the box method: Multiplying binomials using the FOIL pattern F O I L Then combine like terms and write in descending order. Example 3: Find h(x) = f(x) g(x) if Example 4: Use FOIL to multiply (x + 3)(3x + 8) f(x) = (2x + 7) and g(x) = (x 9). Objective #3: Can you multiply polynomials? a) (x 7)(3x + 4) b) (a 5)(a + 3) c) Find h(x) = f(x) g(x) if f(x) = (3x + 2) and g(x) = (3x 2) d) (m + 7)(m 3) + (m 4)(m + 5) 5
Reflect #1: In FOIL, which of the products combine to form the x-term? Which products combine to form the constant term? What about the x²-term? x-term: constant term: x²-term: Finding Special Products of Polynomials SQUARE OF A BINOMIAL *What would you expect the product of (x + 3) 2 to be? Foil it out! Was your prediction correct? *Predict the product of (x 5) 2 Foil it out! Was your prediction correct? Example 5: Find each square of a binomial. a) (x + 4) 2 b) (3x 2) 2 6
Objective #4: Can you square binomials? a) (x + 5) 2 b) (m 8) 2 c) (3c 2) 2 SUM AND DIFFERENCE PATTERN (also called multiplying conjugates) Example 6: Find each product. Do you see a pattern? a) (x + 4)(x 4) b) (2a 7b) (2a + 7b) The Sum and Difference Pattern described using algebra: (a + b)(a b) = 2 2 Objective #5: Can you use the Sum and Difference Pattern to find the product of two binomials? a) (x + 5)(x 5) b) (m 8n)(m + 8n) c) (3c + 2)(3c 2) 7
8.3 Guided Notes: Factor and Solve using Greatest Common Factor (GCF) Warm-Up Simplify the following: 1) 2x(x 5x 3 ) 2) 2 5( 20 4) 3) (x + 3)(x 4) FACTORING out a greatest common factor (Box Method). Step 1: Write the terms the boxes. Step 2: Pull out the greatest common factor to the. Step 3: Write the other product at the tops of the boxes. Step 4: Write your answer as a monomial times a binomial. Example 1: Factor out the greatest common factor (GCF). a) 5x + 20 b) 8x 4x 2 c) 16x + 40y + 8 d) 6x 2 30x 3 Objective #6: Can you factor polynomials using the Greatest Common Factor? a) 4m 2 b) -5x 2 10x c) 6y + 15 d) -9m 3 + m 2 2m 8
Factoring by grouping (the Box Method) Step 1: FACTOR OUT GCF!!! Step 2: Draw a 2x2 box. Step 3: Write your terms in descending order. Step 4: Put one in each section of the box. Step 5: Pull out a Greatest Common Factor (to the.) Step 6: Write your answer as a product of two binomials. Example 2: Factor each expression. a) 6y 3 + 3y 2 + 12y + 6 b) 3x 3 + 6x 2 15x + 30 c) x 4 + 12x 3x 2 + 4x 3 d) 6a 3 3a 2 + 8a 4 Objective #7: Can you use grouping to factor polynomials? a) 10x 3 + 40x 2 20x 80 b) x 3 + 5x 2 2x + 10 c) x 3 4x 2 6x + 24 d) 3x 4 4x 3 + 5x 2 20x 9
Reflect #3: Explain what we are doing when we factor a polynomial. Zero-Product Property Let a and b be real numbers. If ab = 0, then = 0 or = 0. Roots of an equation: Also called Example 3: Solve each equation. a) (x 5)(x + 4) = 0 b) 2a(6a + 1) = 0 Example 4: Solve 2x 2 32x = 0 by factoring. Sketch a graph that includes the x-intercepts. 10
Example 5: Solve 2x 2 8x = 0 by factoring. Sketch a graph that includes the x-intercepts. Objective #8: Can you solve equations by factoring? a) (x + 2)(x 8) = 0 b) 4x 2 + 24x = 0 c) 8p 2 24p = 0 8.4: Factoring and Solving Trinomials and a Difference of Perfect Squares Warm-Up 1) What is the simplified form of (x 6) 2? A. x 2 12x 36 B. x 2 12x + 36 C. x 2 36 D. x 2 + 36 2) What is the product of (x + 3)(x 2 + 2x 4)? 11
Work in groups to FOIL the following: (x + 5)(x 3) (x + 2)(x + 8) (x + 3)(x + 3) (x 5)(x 4) (x 3)(x 10) (x 5)(x + 8) Reflect #4: What are the patterns? How can you short-cut this? Factoring is the opposite of FOILing Example 1: Factor x 2 + 6x + 5 Example 2: Factor each expression. a) x 2 + 10x + 16 b) a 2 5a + 6 c) x 2 + 7x 30 d) b 2 + 10x + 9 e) y 2 y 6 f) x 2 + 2x + 1 g) 3y 2 + 9y 30 h) x 2 + 4x + 12 i) w 3 7w 2 30w 12
Objective #9: Can you factor trinomials in the form x 2 + bx + c? a) 2x 2 + 14x 24 b) x 2 9x + 8 c) y 2 + 4y 21 d) k 3 + 5k 2 50k VOCABULARY: Difference of Two Perfect Squares Factoring the differences of two squares a 2 b 2 = ( + ) ( ) OR use the box method! Example 2: Factor, if possible. a) z 2 81 b) 12x 6 + 27 c) 36a 2 25b 10 d) 2 50n 8 e) b 2 + 100 (be careful!) Objective #10: Can you factor the difference of two squares? a) 2x 2 + 98 b) x 3 81x c) 4a 2 + 64 d) x 2 - y 2 e) 16x 2 4y 2 13
Steps for Solving Quadratic Equations by factoring: 1) Get a on one side of the equation. 2) Factor 3) Use the zero-product property to find the answer(s), which are called,,, or. Example 3: Solve the following equations. Sketch a graph that includes the roots. a) 2x 2 + 14x = 36 b) x 2 + 144 = 0 c) 27b 2 = 72b d) x 2 8x = 16 Objective #11: Can you solve quadratic equations and sketch a graph of the solution? a) x 2 = 196 b) x 2 x = 6 c) 2x 2 22x + 60 = 0 14
8.5 PRACTICE DAY 8.6 Guided Notes: Factoring and Solving Trinomials in the form ax 2 + bx + c Warm-Up 1) Simplify: 2) Simplify: 3) Simplify: (2x + 3)(3x 2 4x + 7) (3x 2 4x + 12) + (2x 2 + 6x 15) (2x 1)(x 5) Factoring trinomials with a leading coefficient other than 1 Step 1: Make a chart to find two integers with a = b and a = ac. Step 2: Draw a 2x2 box. Step 3: Put the terms in your box ( the middle term into 2 terms, look at your chart!) Step 4: Factor. Example 1: Factor 2x 2 llx + 5. + = -11 = 10 Check: Example 2: Factor each expression. a) 3n 2 + 2n - 8 b) 2y 2 13y 7 15
c) 4x 2 + 4x + 3 Objective #12: Can you factor trinomials in the form ax 2 + bx + c? a) 3x 2 5x + 2 b) 2m 2 + m 21 c) 9y 2 + 6y + 1 d) x 2 10x 25 16
Example 3: Solve the following quadratic equations and sketch a graph of the solutions. a) 3x 2 x = 10 b) 6x 2 2x = 4 c) 0 = 4x 2 + 8x + 60 d) 3x 2 28x = 55 17
Objective #13: Can you factor and solve trinomials in the form ax 2 + bx + c? a) 0 = 2x 2 + 5x 3 b) 4x 2 10x = 6 Reflect #6: Compare and contrast your answers and graphs for Objective 13. 8.7 Guided Notes: Factoring Completely Warm-Up 1) Factor: x 2 + 5x + 6 2) Factor: x 2 64 3) Simplify: (x + 7) 2 4) Simplify: (4x 2 3x + 7) (7x 2 6x + 2) 18
Trinomials with More Than One Variable (the Box Method) Be careful when filling out the sections of your box Check your results with the original problem! Do you have all the variables? Example 2: Factor each polynomial. a) x 2 +14xy + 24y 2 b) y 2 10yz + 9z 2 c) 9s 2 + 6st + t 2 Factoring Completely Step 1: If possible, factor out a Greatest Common Factor. Step 2: Can you factor the binomial or trinomial any further? Step 3: Keep factoring until each portion of your answer is fully factored. Example 3: Factor each polynomial, completely. a) 5a 4 405 b) 2x 2 8x 10 19
c) x 4 16 d) x 3 x 2 + 25x + 25 e) 3r 3 21r 2 + 30r f) 9d 4 4d 2 Objective #13: Can you factor completely? a) 2y 4 32 b) 49y 2 25w 6 c) a 2 8ab + 16b 2 d) y 2 + 14yz + 49z 2 20
Objective #14: Can you solve after factoring completely? a) 0 = 45 80m 2 b) 5r 2 20r = 20 d) - x 3 = 2x 2 15x e) 4g 2 + 20g + 24 = 0 21