Chapter 3 notes: Chapter 3-1 Polynomials Obj: SWBAT identify, evaluate, add, and subtract polynomials A monomial is a number, a variable, or a product of numbers and variables with whole number exponents Examples: A polynomial - is a monomial or a sum or difference of monomials (polynomials have no variables in denominators, no roots or absolute values of variables, and all variables have whole number exponents) Determine if the following are Polynomials yes or no explain 3x 4 3 x 7 2z 12 + 9z 3 1 2 x 8 5y 2 x2 + 3x 7 m 3.73 + m 2 1 2 a7 Degree of a monomial is the sum of the exponents of the variables (and only the variables!!) Identify the degree of the following monomials a) x 4 b) a 2 b 4 c) 12 d) 4 2 x 2 y 7 z Degree of a Polynomial is determined by the term with the greatest degree a) x 2 + 7x 4 5 b) 6 + 2x 4x 7 + 3x 5 Standard Form a polynomial with one variable is in Standard Form when its terms are written in descending order by degree. Degree of Polynomial
Ex 5x 3 + 8x 2 + 3x 17 3 2 1 0 Leading Coefficient Degree of each term Classifying Polynomials By Number of Terms and By Degree Polynomials classified by the number of terms: Monomial one term Binomial Trinomial Polynomials classified by degree Name Degree Example Constant 0 Linear 1 Quadratic 2 Cubic 3 Quartic 4 Quintic 5
Classify the following polynomials Rewrite into Standard Form, Identify the Leading Coefficient, Then classify by the Degree and Number of Terms a) 2x + 4x 3 1 b) 7x 3 11x + x 5 2 c) x 2 7 d) 4 9x 2 + 3x 12x 3 + 7x 2 Adding and Subtracting Polynomials Add or subtract, write answer in Standard Form a) (3x 2 + 7x + x) + (14x 3 + 2 + x 2 x) b) ( 36x 2 + 6x 11) + (6x + 16x 3 5) c) (5x 3 + 12 + 6x 2 ) (15x 2 + 3x 2) d) ( 4x + 5x 2 ) ( 4x 5x 2 + 7)
Sum up What is a monomial? A Polynomial? Name some TYPES of Polynomials How do we determine the degree of a Monomial?...Of a Polynomial? Re-write in Standard Form: a) 3 x 3 + 2x 2 b) 3x 2 + 2x 4 + 12x x 5 + 2 What is the leading coefficient, degree from above classify the polynomial by degree and number of terms HW 3.1 pg 154, 10-13 and 19-30 HW 3.1 pg 154, 10-13 and 19-30
Do now pre 3.2 1) ( 2x + 1 ) + (2x + 1) 2) (2x + 1)(2x + 1) 3) ( x + y ) 3
Advanced Algebra 2 3.2 Multiplying polynomials Objective: Multiply polynomials to expand binomial expressions that are raised to positive integers Multiplying Polynomials Pascal s Triangle: (notice the patterns) Binomial Expansion How do you choose which row of Pascal s triangle to use? What is the pattern in the binomial expansion A good trick
Examples of binomial expansion: 1) (y - 3) 4 (this one is tricky) 2) (4z + 5) 3 ( 2x - 3) 5
Group Practice Find the product: 1. 2. 3. 4. Expand the expression 5. 6. ( 3x - 2y) 4 HW: pg. 162-163 20,23,25,27,29,47,54,60
Chapter 3-3 Dividing Polynomials (Day 1) Obj: SWBAT Use Long Division to Divide Polynomials Warm up.divide the following a) x5 b) 12x6 y 4 z 8 x 2 4x 2 y 2 z 3 x2 c) x 5 x3 d) x 3 Method 1: Dividing a Polynomial by a Monomial Divide/Simplify the following 12x 2 y+3x 3x = 12x 2 y 3x + 3x 3x = Break-up as separate fractions You try 3x 2 y+6x 3 y 2 +18xy 3xy
Method 2: Dividing a Polynomial by a Binomial First a walk down memory lane remember long division of numbers? 163 3 It is possible to do the same with polynomials. Simplify x 2 x 30 x 6 Simplify (h 2 11h + 28)(h 4) 1
Simplify (x 4 2x 3 + x 1)/(x + 1) hint: need a placeholder You try.. 8x 2 y 3 28x 3 y 2 4xy 2 c 2 +4c 21 c+7 (m 2 3m 7) (m + 2)
Sum up. Exponent rules x 7 x 3 x 2 x 6 x0 Dividing a polynomial by a monomial 20x 2 10x 5x Dividing a polynomial by a binomial long division Hw: Worksheet
Recall long division. Chapter 3-3 Dividing Polynomials (Day 2) Obj: SWBAT Use Synthetic Division to Divide Polynomials (6x 3 19x 2 + x + 6)(x 3) 1 6x 2 x 2 A simpler process called Synthetic Division has been devised to divide a polynomial by a binomial. Let s use the same problem: Find the a from the Divisor (x 3) a is 3 (6x 3 19x 2 + x + 6) (x 3) (x a) Take the coefficients
Let s try.(x 2 12x 45) (x + 3) (x + 3) is (x 3) so a is -3 More examples a) (2x 2 + x 10) (x 3) b) (x 5 3x 2 20)(x 2) 1 c) (4x 4 5x 2 + 2x + 4) (2x 1) Need to be a coefficient of 1 So divide each ( poly) 2 ( ) 2 Then need to fix at the end
Synthetic Substitution A similar process to synthetic division, synthetic substitution can be used to evaluate a polynomial Example: Evaluate P(x) = x 3 4x 2 + 3x 5 for x = 4 4 1-4 3-5 Check: Plug 4 into the polynomial P(x) = x 3 4x 2 + 3x 5 P(4) = Use synthetic substitution for : a) P(x) = x 3 + 3x 2 + 4 for x = 3 b) P(x) = 4x 4 + 2x 3 + 3x + 5 for x = 1 2 P ( 1 2 ) = 7 2
Sum up We learned a few ways to divide polynomials. Name them. When using synthetic division, what does it mean when the last sum is a zero (no remainder)? When using synthetic division, what does the remainder mean, and how does it relate to synthetic substitution? HW 3-3 (day 2) pg 170, 6-9, 19-24, 25, 26, 31
Adv. Alg 2 Chapter 3.5 Finding Roots of Polynomial Equations (Day 1) Obj: SWBAT Solve Polynomial Equations by Factoring, and Identify the Multiplicity of roots Recall: Solving Quadratics using factoring and the Zero Product Property Solve: 3x 2 6x 24 = 0 look for what, first? Using Factoring to solve Polynomial Equations Solve: a) 2x 3 + 4x 2 30x = 0 GCF?...Re-Factor? b) 3x 5 + 18x 4 + 27x 3 = 0 The MULTICIPLICITY of a root, r, is the number of times that (x r) is a factor of P(x) c) x 4 13x 2 = 36 no GCF but mimics a quadratic in look and factorability
d) 4x 6 + 4x 5 24x 4 = 0 e) 2x 6 10x 5 12x 4 = 0 Sometimes a polynomial equation has a factor that appears more than once. This creates a multiple root. In example b, (3x 5 + 18x 4 + 27x 3 = 0), the polynomial has 2 multiple roots 0 and -3 (as a matter of fact, 0 appeared 3 times and -3, 2 times) Example x 3 9x 2 + 27x 27 = 0 3.5 (day 1) pg 186, 2-7, 15-20
Chapter 3-5 (Day 1) Warm up Factor Completely: 2y 3 + 4y 2 30y Solve the quadratic through ANY method besides factoring 2x 2 12x = 16 Write the simplest polynomial that has 3 2i as a root
Chapter 3-5 Finding Roots of Polynomial Equations (Day 2) Obj: SWBAT use the Rational Root Theorem AND Irrational Root Theorem to solve equations (GOAL - Find ALL the Roots of a polynomial) Translation: IF a Rational Root exists All possible rational roots = Q Example 1: Given p q P(x) = x 3 + 2x 2 x 2, find all POSSIBLE rational roots Now Let s test them to see if any of them work
Example 2: Given P(x) = 2x 3 11x 2 + 12x + 9 find all POSSIBLE rational roots, then identify any roots if possible. p q 3, 3 -½ Example 2: Given P(x) = x 3 5x 2 22x + 56 find all POSSIBLE rational roots, then identify any roots if possible. p (hint: try 2 as a root) q
Translation: If a + b c is a root/zero..then a b c is a root/zero Examples: Identify/list ALL the possible rational roots of the following polynomials, then try to find all the real roots (Rational and irrational) a) x 3 3x 2 2x + 4 p q (Hint start with the LOW roots first) 1 2, 1 ± 5 d) 2x 3 9x 2 + 2 = 0 1 2, 2 ± 6
Chapter 3-6 The Fundamental Theorem of Algebra (Day 1) Obj: SWBAT Identify ALL of the Roots of a Polynomial Equation GOAL: The Goal in this Chapter so far has been to find the roots of a Polynomial Function P(x). WHY: Finding the Roots can help in graphing P(x) or solve equations involving a polynomial function HOW: We have several TOOLS to help find ALL the roots of P(x): -Factoring (and Zero Product Property) -Division (Long, and Synthetic) -Rational Root Theorem ( p q ) - Irrational Root Theorem (If a + b c is a root, then a b c is a root) -When P(x) depressed to a Quadratic Complete the Square and Quadratic Formula. Complex Conjugate Root Theorem (If a + bi is a root, then a bi is a root) -When P(x) depressed to a Quadratic Complete the Square and Quadratic Formula. Real Roots Complex Roots
Given a Function, P(x), and a Zero/Root or a Factor, find ALL the Zeros Ex 1. P(x) = x 3 5x 2 + 11x 15; (x 3) Ex 2. P(x) = x 4 + 5x 2 + 4; 2i is a root
Ex 3. f(x) = x 3 + 5x 2 + x + 5; (x + 5) Ex 4. P(x) = 3x 3 + 7x 2 + 11x + 3; (3x + 1)
Ex 5. f(x) = x 4 4x 3 + 12x 2 + 4x 13; (2 + 3i) (Hint: sum and product of roots) HW: 3.6 D1 worksheet
Chapter 3-6 The Fundamental Theorem of Algebra (Day 2) Obj: SWBAT Write a Polynomial Equation of Least Degree with Given Roots You can write a polynomial function when given the roots! Examples - Write the simplest polynomial function with the following zeros: a) 3 and 5 (EASY we ve done this! 2 DIFFERENT ways) b) 2, 1, and 1 c) 1 + 3, and 5 (hint: use sum and product of roots)
d) 3i, and 7 (Hey What degree should this be?) e) 1 2i and 3 (day 2) pg 193, 1-8 (hints: #5 2 works, #6 3 works)
Chapter 3-7 Investigating Graphs of Polynomial Functions Objective: To use properties of End Behavior to help graph Polynomial Functions The term with the highest power CONTROLS THE ACTION.
Identify the leading coefficient, degree and end behavior a. q(x) = -4x 3-3x 2 +5x + 6 b. p(x) = x 6-7x 5 + x 3 Identify whether the function graphed has an odd or even degree and a positive or negative leading coefficient.
Graph the following function f(x) = x 3 + 3x 2 6x 8 1) Identify possible Rational roots and test them to reduce the polynomial to a Quadratic
2) Graph the following function f(x) = x 3 + 4x 2 + x 6 First: Identify possible Rational roots and test them to reduce the polynomial to a Quadratic Homework p.199/ Check it out - 3a and p. 201/2-9
Chapter 3-8 Transforming Polynomial Functions Obj: To Transform Polynomial Functions Warm Up: Given f(x) = 1 4 (x 3)2 + 5 name the different transformations that have taken place to the parent graph Translating Polynomial Functions Ex 1 For f(x) = x 3 + 4, write the new function, g(x), if the function is moved up 3 units. Then sketch both graphs
Ex 2 For f(x) = x 3 + 4, write the rule, given g(x) = f(x 5). Then sketch both graphs Ex. 3 Reflecting Polynomial Function Let f(x) = x 3 7x 2 + 6x 5. Write a function g that performs each transformation. a) Reflect f(x) across the x-axis b) Reflect f(x) across the y-axis Ex 4 Compressing and Stretching Polynomial Functions Given f(x) = x 4 4x 2 + 2. Write a function g that performs each transtormation. a) Vertically stretch f(x) by a factor of 2 b) Horizontally compress f(x) by a factor of 1 3 c) Vertically compress f(x) by a factor of ½
Ex. 5 Write a function that transforms f(x) = 6x 3 3 in each of the following ways: a) Compress vertically by 1 3 THEN shift 2 units right b) Reflect across the y-axis THEN shift 2 units down Ex. 6 Given f(x) = x 3, write a function that illustrates a vertical compression of 1 followed by 4 a horizontal shift of 5 units right and a vertical shift of 3 units down. Ex. 7 Given f(x) = x 4, write a function that illustrates a reflection across the x-axis, a vertical stretch of 5, followed by a horizontal shift of 2 units left and a vertical shift of 9 units down. HW 3.8 pg 207, 14 24,26,28,31,32,35-37.
Practice B Transforming Polynomial Functions For f(x) x 3 1, write the rule for each function and sketch its graph. 1. g(x) f(x 4) 2. g(x) 3f(x) 3. 1 g( x) f x 2 Let f(x) x 3 4x 2 5x 12. Write a function g(x) that performs each transformation. 4. Reflect f(x) across the y-axis 5. Reflect f (x) across the x-axis Let f(x) x 3 2x 2 3x 6. Describe g(x) as a transformation of f (x) and graph. 6. 1 g( x) f ( x) 4 7. g(x) f (x 6) Write a function that transforms f (x) x 3 4x 2 x 5 in each of the following ways. Support your solution by using a graphing calculator. 8. Move 6 units up and reflect across the y-axis. 9. Compress vertically by a factor of 0.25 and move 3 units right. Solve. 10. The number of participants, N, in a new Internet political forum during each month of the first year can be modeled by N(t) 4t 2 t 2000, where t is the number of months since January. In the second year, the number of forum participants doubled compared to the same month in the previous year. Write a function that describes the number of forum participants in the second year.