Polynomial Functions


 Julius Montgomery
 3 years ago
 Views:
Transcription
1 Polynomial Functions
2 Polynomials A Polynomial in one variable, x, is an expression of the form a n x 0 a 1 x n 1... a n 2 x 2 a n 1 x a n The coefficients represent complex numbers (real or imaginary), ao is not zero, and n represents a nonnegative integer.
3 The degree of a polynomial is a greatest exponent of the variable. The leading coefficient is a coefficient of the variable with the greatest exponent. Zeros of a polynomial function are values of x for which f(x) = 0
4 Example 1 Consider the polynomial function f(x) = x 3  x 27x + 3. a. State the degree and leading coefficient of the polynomial. x 3  x 27x + 3 has a degree of 3 and a leading coefficient of 1.
5 b. Determine whether 3 is a zero of f(x). Evaluate f(x) = x 3  x 27x + 3 for x = 3. That is, find f(3). f(3) = (3) 3  (3) 27(3) + 3 x = 3 f(3) = f(3) = 0 Since f(3) = 0, 3 is a zero of f(x) = x 3  x 27x + 3.
6 Fundamental Thm. Of Algebra Every Polynomial Equation with a degree higher than zero has at least one root in the set of Complex Numbers. COROLLARY: A Polynomial Equation of the form P(x) = 0 of degree n with complex coefficients has exactly n Roots in the set of Complex Numbers. P (x) = k(x r1)(x  r2)(x  r3) (x  rn )
7 Example 2 a. Write a polynomial equation of least degree with roots 3, 2i, and 2i. b. Does the equation have an odd or even degree? How many times does the graph of the related function cross the xaxis?
8 a. If x = 3, then x  3 is a factor of the polynomial. Likewise, if x = 2i and x = 2i, then x  2i and x  (2i) are factors of the polynomial. Therefore, the linear factors for the polynomial are x  3, x  2i, and x + 2i. Now find the products of these factors. (x  3)(x  2i)(x + 2i) = 0 (x  3)(x 24i 2 ) = 0 (x  3)(x 2 + 4) = 04i 2 = 4(1) or 4 x 33x 2 + 4x  12 = 0 A polynomial with roots 3, 2i, and 2i is x 33x 2 + 4x  12 = 0.
9 b. The degree of this equation is 3. Thus, the equation has an odd degree since 3 is an odd number. Since two of the roots are imaginary, the graph will only cross the xaxis once.
10 Example 3 State the number of complex roots of the equation 4x 43x 21 = 0. Then find the roots and graph the related function. Factor the equation to find the roots. 4x 43x 21 = 0 (4x 2 + 1)(x 21) = 0 (4x 2 + 1)(x + 1)(x  1) = 0 To find each root, set each factor equal to zero.
11 (4x 2 + 1)(x + 1)(x  1) = 0 4x 2 + 1= 0 1 x 2 =  4 x = 1 4 x = 1 i 2 x + 1= 0 x  1 = 0 x= 1 x = 1 The roots are 1, 1, and 1. i 2
12 Lesson 4.2 Quadratic Equations.
13
14 Example 1. Solve the equation by completing the square. x 24x  5= 0 x 24x = 5 x 24x + 4= (x  2) 2 = 9 x 2 = 3 b 4 4 ( ) x  2= 3 or x  2= 3 x= 5 x= 1 Answer: The roots of the equation are 1 and 5.
15 Example 2. Solve 3x 2 + 4x + 4 = 0 by completing the square. b 4 ( )
16
17
18
19 Example 3. Find the discriminant of x 2 + 4x + 2 = 0 and describe the nature of the roots of the equation. Then solve the equation by using the Quadratic Formula. The discriminant b 24ac = 4 24(1)(2) = 8 Since the value of the discriminant is greater than zero, there are two real rational roots.
20
21 Example 4. Solve 2x 2 + 3x + 4 = 0.
22 Lesson 4.3 The Remainder and Factor Theorems
23
24 Example 1. Synthetic Substitution Divide f(x) = 2x 4 5x 2 + 8x 7 by x6 using synthetic division Notice that there is no x 3 term. A zero is placed in this position as a placeholder. Answer: The quotient is is R is 2453.
25 Example 2. Divide f(x) = 2x 4 5x 2 + 8x 7 by x6 using direct substitution. State whether the binomial is a factor of the polynomial. Replace x with 6. Original function Replace x with 6. Simplify. Answer: f(6) is not a zero. So, x 6 is not a factor of the polynomial.
26
27 Example 3. Determine the binomial factors of x 3 + 4x 2 15x 18. One root is x = 3 and the factor is (x  3). Use synthetic division to find the rest of the factors.
28 Example 3. Use the Factor Theorem Determine the binomial factors of x 3 + 4x 2 15x 18. Because f(3) = 0, x  3 is a factor. Find the depressed polynomial
29 The polynomial x 2 + 7x + 6 is the depressed polynomial. Check to see if this polynomial can be factored. x 2 + 7x + 6 = (x + 6)(x + 1) Factor the trinomial. Answer: So, x 3 + 4x 2 15x 18 = (x 3)(x + 6)(x + 1).
30 Example 4. Find the value of k so that the remainder of (3x 4 + 8x 32x 2  kx + 4) (x + 2) is 0. If the remainder is to be 0, x + 2 must be a factor of 3x 4 + 8x 32x 2  kx + 4. So, f(2) must equal 0. f(x)= 3x 4 + 8x 32x 2  kx + 4 f(2)= 3(2) 4 + 8(2) 32(2) 2  k(2) = k + 4 Replace f(2) with 0. 0 = 2k = k
31 The value of k is 10. Check using synthetic division
32 Lesson 4.4 The Rational Root Theorem
33
34 Example 1. List the possible rational roots of 1x 3  x 210x  8 = 0. Then determine the rational roots. possible values of p: 1, 2, 4, 8 possible values of q: 1 q p Answer: the possible rational roots p/q: 1, 2, 4, 8
35 You can use a graphing utility to narrow down the possibilities. You know that all possible rational roots fall in the domain 8 x 8. So, set your xaxis viewing window at [9, 9]. Graph the related function f(x) = x 3  x 210x  8. A zero appears to occur at 2. Use synthetic division to check that 2 is a zero.
36 Solve by factoring x 23x  4 =0 (x  4)(x + 1)=0 x = 4, x = 1 Answer: 2, 1, and 4
37
38 Example 2. Find Numbers of Positive and Negative Zeros Find the number of possible positive real zeros and the number of possible negative real zeros for x 45x Then determine the rational zeros. To determine the number of possible positive real zeros, count the sign changes for the coefficients. There are two changes. So, there are 2 or 0 positive real zeros.
39 To determine the number of possible negative real zeros, find f(x) and count the number of sign changes There are two changes. So, there are 2 or 0 negative real zeros.
40 Example 2. f(x) = x 45x 2 +4 possible values of p: 1, 2, 4 possible values of q: 1 Determine the rational zeros. Test the possible zeros using the Remainder Theorem. f(1) = 0, f(1) = 0, f(2) = 0, and f(2) = 0 Answer: 1, 1, 2, and 2
41
42 Suppose y = f(x) represents a polynomial function with real coefficients. If a and b are two numbers with f(a) negative and f(b) positive, the function has at least one real zero between a and b.
43 Example 1: Determine between which consecutive integers the real zeros of f(x) = x 3 + 3x 24x + 6 are located. Use the TABLE feature. The change in sign between 5 and 4 indicates that a zero exists between 5 and 4.
44 Example 2 Approximate the real zeros of f(x) = 5x 32x 24x + 1 to the nearest tenth. There are three complex zeros for this function. According to Descartes Rule of Signs, there are two or zero positive real roots and one negative real root. Use the TABLE feature of a graphing calculator. To find the zeros to the nearest tenth, use the TBLSET feature changing Tbl to 0.1. There are zeros between 0.9 and 0.8, between 0.2 and 0.3, and at 1.
45 Since 0.36 is closer to zero than , the zero is about Since 0.16 is closer to zero than , the zero is about 0.2.
46 The third zero occurs at 1.
47 Upper Bound Theorem Suppose c is a positive number and P(x) is divided by x c. If the resulting quotient and remainder have no change in sign, then P(x) has no real zero greater than c. Thus c is an upper bound of the zeros of P(x). Lower Bound Theorem If c is an upper bound of the zeros of P(x), then c is a lower bound of the zeros of P(x).
48 Example 3 Use the Upper Bound Theorem to find an integral upper bound and the Lower Bound Theorem to find an integral lower bound of the zeros of f(x) = x 3 + 5x 22x  8. The Rational Root Theorem tells us that 1, 2, 4, and 8 might be roots of the polynomial equation x 3 + 5x 22x  8 = 0. These possible zeros of the function are good starting places for finding an upper bound. f(x) = x 3 + 5x 22x  8 f(x) = x 3 + 5x 2 + 2x  8 r r An upper bound is Since 6 is an upper bound of f(x), 6 is a lower bound of f(x). This means that all real zeros of f(x) can be found in the interval 6 x 2.
49 Lesson 4.6
50 The best way to solve a rational equation: This can be done by multiplying each side of the equation by the LCD.
51 Example 1. Solve for x What is the LCD? (x 2)(x 5) 7 x x 2 6 x 5 (x+2)(x5) x 5 7(x 5) 6(x 2) 7x 35 6x 12 x x 47 (x 2)(x 5)
52 The other method of solving rational equations is crossmultiplication. 7 x 2 6 x 5 7 (x 5) 6 (x 2) 7x 35 6x 12 x x 47
53 Example 2. Solve for x Step 1: Find the LCD x 1 3x 6 5x 6 x 1 2 Hint: Factor the denominator x 1 3x 6 Therefore. This denominator can be factored into 3(x2) LCD 6(x 2)
54 Step 2: Multiply both sides of equation by LCD. This eliminates the fraction. 6(x 2) x 1 3(x 2) 6(x 2) 5x 6 6(x 2) x 1 2 2(x 1) (x 2)5x 6
55 2(x 1) 5x(x 2) 1 6 2x 2 5x 2 10x 6 2x 2 2x 2 0 5x 2 12x 4 0 (5x 2)(x 2) x 2 5 x 2
56
57 Example 3 Decompose x 2 x 9 3x 10 into partial fractions. First factor the denominator. x 23x  10= (x  5)(x + 2) Express the factored form as the sum of two fractions using A and B as numerators and the factors as denominators. x 2 x 9 3x 10 x A 5 x B 2
58 Eliminate the denominators bymultiplying each side by the LCD, (x  5)(x + 2). (x 5)(x 2) 2 x x 9 3x 10 (x 5)(x 2) x A 5 (x 5)(x 2) x B 2 x + 9= A(x + 2) + B(x  5) Eliminate B by letting x = 5 so that x  5 becomes 0. x + 9= A(x + 2) + B(x  5) 5 + 9= A(5 + 2) + B(55) 14= 7A 2= A
59 x + 9= A(x + 2) + B(x  5) Eliminate A by letting x = 2 so that x + 2 becomes 0. x + 9= A(x + 2) + B(x  5) = A(2 + 2) + B(25) 7= 7B 1= B Now substitute the values for A and B to determine the partial fractions. A x 5 x B 2 x x 2
60 x 2 x 9 3x 10 x x 2
61
62 Example 4: Solve (x 1)(x 3) 0 2 (x 2)(x 4) 1. Solve the numerator (x +1)(x 3) = 0 x = 1 and x = 3 2. Solve the denominator (x  2)(x +4)(x + 4) = 0 x = 2 and x = Make vertical dashed lines through 4, 1, 2, 3
63 4. Test a convenient value within each interval in the original rational inequality to see if the test value is a solution. For x <  4, test x = 5: (x 1)(x 3) (x 2)(x 4) 2 0 ( 5 ( 5 1)( 5 2)( 5 3) 4) 2 ( )( ) ( )( ) So, x < 4 is not a solution.
64 For 4 < x < 1, test x = 2.5: (x 1)(x 3) (x 2)(x 4) 2 0 ( 2.5 ( 2.5 1)( 2.5 2)( 2.5 3) 4) 2 ( )( ) ( )( ) So,  4 < x < 1 is not a solution.
65 For 1 < x < 2, test x = 0: (x 1)(x 3) (x 2)(x 4) 2 0 (0 1)(0 3) (0 2)(0 4) 2 ( )( ) ( )( ) So, 1 < x < 2 is a solution.
66 For 2 < x < 3, test x = 2.5: (x 1)(x 3) (x 2)(x 4) 2 0 (2.5 1)(2.5 3) (2.5 2)(2.5 4) 2 ( )( ) ( )( ) So, 2 < x < 3 is not a solution.
67 For x > 3, test x = 4: (x 1)(x 3) (x 2)(x 4) 2 0 (4 1)(4 3) (4 2)(4 4) 2 ( )( ) ( )( ) So, x > 3 is a solution.
68 This solution can be graphed on a number line.
69 Lesson 4.7
70 Equations in which radical expressions include variables are To solve radical equations 1. Isolate the radical on one side of the equation 2. Raise each side of the equation to the proper power to eliminate the radical expression.
71 Example 1. Solve
72
73 Example 2. Solve 50 7x 8 x 50 7x x x x 2 16x 64 0 x 2 16x 64 7x 50 0 x 2 9x 14 0 x ( x 2)( x 7 2 x ) 7
74 Check both solutions to make sure they are not extraneous.
75 Example 3. Solve
76 Check both solutions to make sure they are not extraneous.
77 Example 4. Solve 5x Solve 5x + 4 = 0, x = Solve 5x 4 8 5x 5x x
78 3. On a number line, mark 0.8 and 12 with vertical dashed lines. For x 0.8, test x = 1: This statement is meaningless. x 0.8 is not a solution.
79 For 0.8 x 12, test x = 0: 0.8 x 12 is a solution.
80 For x 12, test x = 13: x 12 is not a solution.
81
82
Never leave a NEGATIVE EXPONENT or a ZERO EXPONENT in an answer in simplest form!!!!!
1 ICM Unit 0 Algebra Rules Lesson 1 Rules of Exponents RULE EXAMPLE EXPLANANTION a m a n = a m+n A) x x 6 = B) x 4 y 8 x 3 yz = When multiplying with like bases, keep the base and add the exponents. a
More informationChapter 8. Exploring Polynomial Functions. Jennifer Huss
Chapter 8 Exploring Polynomial Functions Jennifer Huss 81 Polynomial Functions The degree of a polynomial is determined by the greatest exponent when there is only one variable (x) in the polynomial Polynomial
More informationNAME DATE PERIOD. Power and Radical Functions. New Vocabulary Fill in the blank with the correct term. positive integer.
21 Power and Radical Functions What You ll Learn Scan Lesson 21. Predict two things that you expect to learn based on the headings and Key Concept box. 1. 2. Lesson 21 Active Vocabulary extraneous solution
More informationA Partial List of Topics: Math Spring 2009
A Partial List of Topics: Math 112  Spring 2009 This is a partial compilation of a majority of the topics covered this semester and may not include everything which might appear on the exam. The purpose
More informationEquations. Rational Equations. Example. 2 x. a b c 2a. Examine each denominator to find values that would cause the denominator to equal zero
Solving Other Types of Equations Rational Equations Examine each denominator to find values that would cause the denominator to equal zero Multiply each term by the LCD or If two terms crossmultiply Solve,
More informationRoots & Zeros of Polynomials. How the roots, solutions, zeros, xintercepts and factors of a polynomial function are related.
Roots & Zeros of Polynomials How the roots, solutions, zeros, xintercepts and factors of a polynomial function are related. A number a is a zero or root of a function y = f (x) if and only if f (a) =
More informationPractice Test  Chapter 2
Graph and analyze each function. Describe the domain, range, intercepts, end behavior, continuity, and where the function is increasing or decreasing. 1. f (x) = 0.25x 3 Evaluate the function for several
More informationChapter 3: Polynomial and Rational Functions
Chapter 3: Polynomial and Rational Functions 3.1 Polynomial Functions A polynomial on degree n is a function of the form P(x) = a n x n + a n 1 x n 1 + + a 1 x 1 + a 0, where n is a nonnegative integer
More informationMAT116 Final Review Session Chapter 3: Polynomial and Rational Functions
MAT116 Final Review Session Chapter 3: Polynomial and Rational Functions Quadratic Function A quadratic function is defined by a quadratic or seconddegree polynomial. Standard Form f x = ax 2 + bx + c,
More informationComplex Numbers: Definition: A complex number is a number of the form: z = a + bi where a, b are real numbers and i is a symbol with the property: i
Complex Numbers: Definition: A complex number is a number of the form: z = a + bi where a, b are real numbers and i is a symbol with the property: i 2 = 1 Sometimes we like to think of i = 1 We can treat
More informationBeginning Algebra. 1. Review of PreAlgebra 1.1 Review of Integers 1.2 Review of Fractions
1. Review of PreAlgebra 1.1 Review of Integers 1.2 Review of Fractions Beginning Algebra 1.3 Review of Decimal Numbers and Square Roots 1.4 Review of Percents 1.5 Real Number System 1.6 Translations:
More informationName: 6.4 Polynomial Functions. Polynomial in One Variable
Name: 6.4 Polynomial Functions Polynomial Functions: The expression 3r 2 3r + 1 is a in one variable since it only contains variable, r. KEY CONCEPT Polynomial in One Variable Words A polynomial of degree
More informationAlgebra I Unit Report Summary
Algebra I Unit Report Summary No. Objective Code NCTM Standards Objective Title Real Numbers and Variables Unit  ( Ascend Default unit) 1. A01_01_01 HAB.1 Word Phrases As Algebraic Expressions 2. A01_01_02
More informationPolynomial and Rational Functions. Copyright Cengage Learning. All rights reserved.
2 Polynomial and Rational Functions Copyright Cengage Learning. All rights reserved. 2.3 Real Zeros of Polynomial Functions Copyright Cengage Learning. All rights reserved. What You Should Learn Use long
More informationSkills Practice Skills Practice for Lesson 10.1
Skills Practice Skills Practice for Lesson.1 Name Date Higher Order Polynomials and Factoring Roots of Polynomial Equations Problem Set Solve each polynomial equation using factoring. Then check your solution(s).
More information3.4. ZEROS OF POLYNOMIAL FUNCTIONS
3.4. ZEROS OF POLYNOMIAL FUNCTIONS What You Should Learn Use the Fundamental Theorem of Algebra to determine the number of zeros of polynomial functions. Find rational zeros of polynomial functions. Find
More information6x 3 12x 2 7x 2 +16x 7x 2 +14x 2x 4
2.3 Real Zeros of Polynomial Functions Name: Precalculus. Date: Block: 1. Long Division of Polynomials. We have factored polynomials of degree 2 and some specific types of polynomials of degree 3 using
More informationSolving Quadratic Equations Review
Math III Unit 2: Polynomials Notes 21 Quadratic Equations Solving Quadratic Equations Review Name: Date: Period: Some quadratic equations can be solved by. Others can be solved just by using. ANY quadratic
More informationUnit 1: Polynomial Functions SuggestedTime:14 hours
Unit 1: Polynomial Functions SuggestedTime:14 hours (Chapter 3 of the text) Prerequisite Skills Do the following: #1,3,4,5, 6a)c)d)f), 7a)b)c),8a)b), 9 Polynomial Functions A polynomial function is an
More informationChapter Five Notes N P U2C5
Chapter Five Notes N P UC5 Name Period Section 5.: Linear and Quadratic Functions with Modeling In every math class you have had since algebra you have worked with equations. Most of those equations have
More informationA repeated root is a root that occurs more than once in a polynomial function.
Unit 2A, Lesson 3.3 Finding Zeros Synthetic division, along with your knowledge of end behavior and turning points, can be used to identify the xintercepts of a polynomial function. This information allows
More informationAlgebra 2 Segment 1 Lesson Summary Notes
Algebra 2 Segment 1 Lesson Summary Notes For each lesson: Read through the LESSON SUMMARY which is located. Read and work through every page in the LESSON. Try each PRACTICE problem and write down the
More informationChapter 2 Polynomial and Rational Functions
Chapter 2 Polynomial and Rational Functions Section 1 Section 2 Section 3 Section 4 Section 5 Section 6 Section 7 Quadratic Functions Polynomial Functions of Higher Degree Real Zeros of Polynomial Functions
More informationAlgebra 2 Honors: Final Exam Review
Name: Class: Date: Algebra 2 Honors: Final Exam Review Directions: You may write on this review packet. Remember that this packet is similar to the questions that you will have on your final exam. Attempt
More informationChapter 2. Polynomial and Rational Functions. 2.5 Zeros of Polynomial Functions
Chapter 2 Polynomial and Rational Functions 2.5 Zeros of Polynomial Functions 1 / 33 23 Chapter 2 Homework 2.5 p335 6, 8, 10, 12, 16, 20, 24, 28, 32, 34, 38, 42, 46, 50, 52 2 / 33 23 3 / 33 23 Objectives:
More informationReading Mathematical Expressions & Arithmetic Operations Expression Reads Note
Math 001  Term 171 Reading Mathematical Expressions & Arithmetic Operations Expression Reads Note x A x belongs to A,x is in A Between an element and a set. A B A is a subset of B Between two sets. φ
More informationReference Material /Formulas for PreCalculus CP/ H Summer Packet
Reference Material /Formulas for PreCalculus CP/ H Summer Packet Week # 1 Order of Operations Step 1 Evaluate expressions inside grouping symbols. Order of Step 2 Evaluate all powers. Operations Step
More informationSection 8.3 Partial Fraction Decomposition
Section 8.6 Lecture Notes Page 1 of 10 Section 8.3 Partial Fraction Decomposition Partial fraction decomposition involves decomposing a rational function, or reversing the process of combining two or more
More informationChapter 2 Formulas and Definitions:
Chapter 2 Formulas and Definitions: (from 2.1) Definition of Polynomial Function: Let n be a nonnegative integer and let a n,a n 1,...,a 2,a 1,a 0 be real numbers with a n 0. The function given by f (x)
More informationx 2 + 6x 18 x + 2 Name: Class: Date: 1. Find the coordinates of the local extreme of the function y = x 2 4 x.
1. Find the coordinates of the local extreme of the function y = x 2 4 x. 2. How many local maxima and minima does the polynomial y = 8 x 2 + 7 x + 7 have? 3. How many local maxima and minima does the
More informationModeling Data. 27 will get new packet. 24 Mixed Practice 3 Binomial Theorem. 23 Fundamental Theorem March 2
Name: Period: PreCal AB: Unit 1: Polynomials Monday Tuesday Block Friday 11/1 1 Unit 1 TEST Function Operations and Finding Inverses 16 17 18/19 0 NO SCHOOL Polynomial Division Roots, Factors, Zeros and
More informationPower and Polynomial Functions. College Algebra
Power and Polynomial Functions College Algebra Power Function A power function is a function that can be represented in the form f x = kx % where k and p are real numbers, and k is known as the coefficient.
More informationPolynomials. Exponents. End Behavior. Writing. Solving Factoring. Graphing. End Behavior. Polynomial Notes. Synthetic Division.
Polynomials Polynomials 1. P 1: Exponents 2. P 2: Factoring Polynomials 3. P 3: End Behavior 4. P 4: Fundamental Theorem of Algebra Writing real root x= 10 or (x+10) local maximum Exponents real root x=10
More informationAlgebra III Chapter 2 Note Packet. Section 2.1: Polynomial Functions
Algebra III Chapter 2 Note Packet Name Essential Question: Section 2.1: Polynomial Functions Polynomials Have nonnegative exponents Variables ONLY in General Form n ax + a x +... + ax + ax+ a n n 1
More information3 Polynomial and Rational Functions
3 Polynomial and Rational Functions 3.1 Polynomial Functions and their Graphs So far, we have learned how to graph polynomials of degree 0, 1, and. Degree 0 polynomial functions are things like f(x) =,
More informationB. Complex number have a Real part and an Imaginary part. 1. written as a + bi some Examples: 2+3i; 7+0i; 0+5i
Section 11.8 Complex Numbers I. The Complex Number system A. The number i = 1 1. 9 and 24 B. Complex number have a Real part and an Imaginary part II. Powers of i 1. written as a + bi some Examples: 2+3i;
More information3.3 Real Zeros of Polynomial Functions
71_00.qxp 12/27/06 1:25 PM Page 276 276 Chapter Polynomial and Rational Functions. Real Zeros of Polynomial Functions Long Division of Polynomials Consider the graph of f x 6x 19x 2 16x 4. Notice in Figure.2
More informationPractice Test  Chapter 2
Graph and analyze each function. Describe the domain, range, intercepts, end behavior, continuity, and where the function is increasing or decreasing. 1. f (x) = 0.25x 3 Evaluate the function for several
More informationWe say that a polynomial is in the standard form if it is written in the order of decreasing exponents of x. Operations on polynomials:
R.4 Polynomials in one variable A monomial: an algebraic expression of the form ax n, where a is a real number, x is a variable and n is a nonnegative integer. : x,, 7 A binomial is the sum (or difference)
More informationAlgebra Summer Review Packet
Name: Algebra Summer Review Packet About Algebra 1: Algebra 1 teaches students to think, reason, and communicate mathematically. Students use variables to determine solutions to real world problems. Skills
More informationChapter 2 Polynomial and Rational Functions
Chapter 2 Polynomial and Rational Functions Overview: 2.2 Polynomial Functions of Higher Degree 2.3 Real Zeros of Polynomial Functions 2.4 Complex Numbers 2.5 The Fundamental Theorem of Algebra 2.6 Rational
More informationCourse Name: MAT 135 Spring 2017 Master Course Code: N/A. ALEKS Course: Intermediate Algebra Instructor: Master Templates
Course Name: MAT 135 Spring 2017 Master Course Code: N/A ALEKS Course: Intermediate Algebra Instructor: Master Templates Course Dates: Begin: 01/15/2017 End: 05/31/2017 Course Content: 279 Topics (207
More informationAn equation is a statement that states that two expressions are equal. For example:
Section 0.1: Linear Equations Solving linear equation in one variable: An equation is a statement that states that two expressions are equal. For example: (1) 513 (2) 16 (3) 4252 (4) 64153 To solve the
More informationEquations and Inequalities. College Algebra
Equations and Inequalities College Algebra Radical Equations Radical Equations: are equations that contain variables in the radicand How to Solve a Radical Equation: 1. Isolate the radical expression on
More informationKing Fahd University of Petroleum and Minerals PrepYear Math Program Math Term 161 Recitation (R1, R2)
Math 001  Term 161 Recitation (R1, R) Question 1: How many rational and irrational numbers are possible between 0 and 1? (a) 1 (b) Finite (c) 0 (d) Infinite (e) Question : A will contain how many elements
More informationSolving Quadratic Equations by Formula
Algebra Unit: 05 Lesson: 0 Complex Numbers All the quadratic equations solved to this point have had two real solutions or roots. In some cases, solutions involved a double root, but there were always
More informationReview all the activities leading to Midterm 3. Review all the problems in the previous online homework sets (8+9+10).
MA109, Activity 34: Review (Sections 3.6+3.7+4.1+4.2+4.3) Date: Objective: Additional Assignments: To prepare for Midterm 3, make sure that you can solve the types of problems listed in Activities 33 and
More informationMath 3 Variable Manipulation Part 3 Polynomials A
Math 3 Variable Manipulation Part 3 Polynomials A 1 MATH 1 & 2 REVIEW: VOCABULARY Constant: A term that does not have a variable is called a constant. Example: the number 5 is a constant because it does
More informationTheorems About Roots of Polynomial Equations. Theorem Rational Root Theorem
 Theorems About Roots of Polynomial Equations Content Standards N.CN.7 Solve quadratic equations with real coefﬁcients that have complex solutions. Also N.CN.8 Objectives To solve equations using the
More informationOBJECTIVES UNIT 1. Lesson 1.0
OBJECTIVES UNIT 1 Lesson 1.0 1. Define "set," "element," "finite set," and "infinite set," "empty set," and "null set" and give two examples of each term. 2. Define "subset," "universal set," and "disjoint
More informationControlling the Population
Lesson.1 Skills Practice Name Date Controlling the Population Adding and Subtracting Polynomials Vocabulary Match each definition with its corresponding term. 1. polynomial a. a polynomial with only 1
More informationLesson 7.1 Polynomial Degree and Finite Differences
Lesson 7.1 Polynomial Degree and Finite Differences 1. Identify the degree of each polynomial. a. 3x 4 2x 3 3x 2 x 7 b. x 1 c. 0.2x 1.x 2 3.2x 3 d. 20 16x 2 20x e. x x 2 x 3 x 4 x f. x 2 6x 2x 6 3x 4 8
More informationCommon Core Algebra 2 Review Session 1
Common Core Algebra 2 Review Session 1 NAME Date 1. Which of the following is algebraically equivalent to the sum of 4x 2 8x + 7 and 3x 2 2x 5? (1) 7x 2 10x + 2 (2) 7x 2 6x 12 (3) 7x 4 10x 2 + 2 (4) 12x
More informationMATH98 Intermediate Algebra Practice Test Form A
MATH98 Intermediate Algebra Practice Test Form A MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Solve the equation. 1) (y  4)  (y + ) = 3y 1) A)
More informationSection 4.1: Polynomial Functions and Models
Section 4.1: Polynomial Functions and Models Learning Objectives: 1. Identify Polynomial Functions and Their Degree 2. Graph Polynomial Functions Using Transformations 3. Identify the Real Zeros of a Polynomial
More informationUnit 1 Vocabulary. A function that contains 1 or more or terms. The variables may be to any nonnegative power.
MODULE 1 1 Polynomial A function that contains 1 or more or terms. The variables may be to any nonnegative power. 1 Modeling Mathematical modeling is the process of using, and to represent real world
More informationCh 7 Summary  POLYNOMIAL FUNCTIONS
Ch 7 Summary  POLYNOMIAL FUNCTIONS 1. An opentop box is to be made by cutting congruent squares of side length x from the corners of a 8.5 by 11inch sheet of cardboard and bending up the sides. a)
More informationPolynomial and Rational Functions. Chapter 3
Polynomial and Rational Functions Chapter 3 Quadratic Functions and Models Section 3.1 Quadratic Functions Quadratic function: Function of the form f(x) = ax 2 + bx + c (a, b and c real numbers, a 0) 30
More informationRon Paul Curriculum Mathematics 8 Lesson List
Ron Paul Curriculum Mathematics 8 Lesson List 1 Introduction 2 Algebraic Addition 3 Algebraic Subtraction 4 Algebraic Multiplication 5 Week 1 Review 6 Algebraic Division 7 Powers and Exponents 8 Order
More information( 3) ( ) ( ) ( ) ( ) ( )
81 Instruction: Determining the Possible Rational Roots using the Rational Root Theorem Consider the theorem stated below. Rational Root Theorem: If the rational number b / c, in lowest terms, is a root
More information5.3. Polynomials and Polynomial Functions
5.3 Polynomials and Polynomial Functions Polynomial Vocabulary Term a number or a product of a number and variables raised to powers Coefficient numerical factor of a term Constant term which is only a
More informationExtra Polynomial & Rational Practice!
Extra Polynomial & Rational Practice! EPRP p1 1. Graph these polynomial functions. Label all intercepts and describe the end behavior. 3 a. P(x = x x 1x. b. P(x = x x x.. Use polynomial long division.
More information= (Type exponential notation with positive exponents)
1. Subtract. Simplify by collecting like radical terms if possible. 2 2 = (Simplify your answer) 2. Add. Simplify if possible. = (Simplify your answer) 3. Divide and simplify. = (Type exponential notation
More informationQuadratic Functions. Key Terms. Slide 1 / 200. Slide 2 / 200. Slide 3 / 200. Table of Contents
Slide 1 / 200 Quadratic Functions Table of Contents Key Terms Identify Quadratic Functions Explain Characteristics of Quadratic Functions Solve Quadratic Equations by Graphing Solve Quadratic Equations
More informationQuadratic Functions. Key Terms. Slide 2 / 200. Slide 1 / 200. Slide 3 / 200. Slide 4 / 200. Slide 6 / 200. Slide 5 / 200.
Slide 1 / 200 Quadratic Functions Slide 2 / 200 Table of Contents Key Terms Identify Quadratic Functions Explain Characteristics of Quadratic Functions Solve Quadratic Equations by Graphing Solve Quadratic
More informationSlide 1 / 200. Quadratic Functions
Slide 1 / 200 Quadratic Functions Key Terms Slide 2 / 200 Table of Contents Identify Quadratic Functions Explain Characteristics of Quadratic Functions Solve Quadratic Equations by Graphing Solve Quadratic
More informationScope and Sequence Mathematics Algebra 2 400
Scope and Sequence Mathematics Algebra 2 400 Description : Students will study real numbers, complex numbers, functions, exponents, logarithms, graphs, variation, systems of equations and inequalities,
More informationMATH98 Intermediate Algebra Practice Test Form B
MATH98 Intermediate Algebra Practice Test Form B MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Solve the equation. 1) (y  4)  (y + 9) = y 1) 
More informationUsing Properties of Exponents
6.1 Using Properties of Exponents Goals p Use properties of exponents to evaluate and simplify expressions involving powers. p Use exponents and scientific notation to solve reallife problems. VOCABULARY
More informationChapter 7 Polynomial Functions. Factoring Review. We will talk about 3 Types: ALWAYS FACTOR OUT FIRST! Ex 2: Factor x x + 64
Chapter 7 Polynomial Functions Factoring Review We will talk about 3 Types: 1. 2. 3. ALWAYS FACTOR OUT FIRST! Ex 1: Factor x 2 + 5x + 6 Ex 2: Factor x 2 + 16x + 64 Ex 3: Factor 4x 2 + 6x 18 Ex 4: Factor
More informationStudy Guide for Math 095
Study Guide for Math 095 David G. Radcliffe November 7, 1994 1 The Real Number System Writing a fraction in lowest terms. 1. Find the largest number that will divide into both the numerator and the denominator.
More informationOctober 28, S4.4 Theorems about Zeros of Polynomial Functions
MAT 171 Precalculus Algebra Dr. Claude Moore Cape Fear Community College CHAPTER 4: Polynomial and Rational Functions 4.1 Polynomial Functions and Models 4.2 Graphing Polynomial Functions 4.3 Polynomial
More information6.1 Using Properties of Exponents 1. Use properties of exponents to evaluate and simplify expressions involving powers. Product of Powers Property
6.1 Using Properties of Exponents Objectives 1. Use properties of exponents to evaluate and simplify expressions involving powers. 2. Use exponents and scientific notation to solve real life problems.
More informationDay 6: 6.4 Solving Polynomial Equations Warm Up: Factor. 1. x 22x x 29x x 2 + 6x + 5
Day 6: 6.4 Solving Polynomial Equations Warm Up: Factor. 1. x 22x  15 2. x 29x + 14 3. x 2 + 6x + 5 Solving Equations by Factoring Recall the factoring pattern: Difference of Squares:...... Note: There
More informationHONORS GEOMETRY Summer Skills Set
HONORS GEOMETRY Summer Skills Set Algebra Concepts Adding and Subtracting Rational Numbers To add or subtract fractions with the same denominator, add or subtract the numerators and write the sum or difference
More information2.1 Quadratic Functions
Date:.1 Quadratic Functions Precalculus Notes: Unit Polynomial Functions Objective: The student will sketch the graph of a quadratic equation. The student will write the equation of a quadratic function.
More information3 What is the degree of the polynomial function that generates the data shown below?
hapter 04 Test Name: ate: 1 For the polynomial function, describe the end behavior of its graph. The leading term is down. The leading term is and down.. Since n is 1 and a is positive, the end behavior
More informationMaintaining Mathematical Proficiency
Chapter Maintaining Mathematical Proficiency Simplify the expression. 1. 8x 9x 2. 25r 5 7r r + 3. 3 ( 3x 5) + + x. 3y ( 2y 5) + 11 5. 3( h 7) 7( 10 h) 2 2 +. 5 8x + 5x + 8x Find the volume or surface area
More informationProcedure for Graphing Polynomial Functions
Procedure for Graphing Polynomial Functions P(x) = a nx n + a n1x n1 + + a 1x + a 0 To graph P(x): As an example, we will examine the following polynomial function: P(x) = 2x 3 3x 2 23x + 12 1. Determine
More informationTABLE OF CONTENTS. Introduction to Finish Line Indiana Math 10. UNIT 1: Number Sense, Expressions, and Computation. Real Numbers
TABLE OF CONTENTS Introduction to Finish Line Indiana Math 10 UNIT 1: Number Sense, Expressions, and Computation LESSON 1 8.NS.1, 8.NS.2, A1.RNE.1, A1.RNE.2 LESSON 2 8.NS.3, 8.NS.4 LESSON 3 A1.RNE.3 LESSON
More informationAlgebra 2 Honors Final Exam StudyGuide
Name: Score: 0 / 80 points (0%) Algebra 2 Honors Final Exam StudyGuide Multiple Choice Identify the choice that best completes the statement or answers the question. 1. Simplify. 2. D Multiply the numerator
More informationSecondary Math 2H Unit 3 Notes: Factoring and Solving Quadratics
Secondary Math H Unit 3 Notes: Factoring and Solving Quadratics 3.1 Factoring out the Greatest Common Factor (GCF) Factoring: The reverse of multiplying. It means figuring out what you would multiply together
More informationNAME DATE PERIOD. Operations with Polynomials. Review Vocabulary Evaluate each expression. (Lesson 11) 3a 2 b 4, given a = 3, b = 2
51 Operations with Polynomials What You ll Learn Skim the lesson. Predict two things that you expect to learn based on the headings and the Key Concept box. 1. Active Vocabulary 2. Review Vocabulary Evaluate
More informationPolynomials: Adding, Subtracting, & Multiplying (5.1 & 5.2)
Polynomials: Adding, Subtracting, & Multiplying (5.1 & 5.) Determine if the following functions are polynomials. If so, identify the degree, leading coefficient, and type of polynomial 5 3 1. f ( x) =
More information171S4.4 Theorems about Zeros of Polynomial Functions. March 27, 2012
MAT 171 Precalculus Algebra Dr. Claude Moore Cape Fear Community College CHAPTER 4: Polynomial and Rational Functions 4.1 Polynomial Functions and Models 4.2 Graphing Polynomial Functions 4.3 Polynomial
More informationChapter R  Basic Algebra Operations (94 topics, no due date)
Course Name: Math 00024 Course Code: N/A ALEKS Course: College Algebra Instructor: Master Templates Course Dates: Begin: 08/15/2014 End: 08/15/2015 Course Content: 207 topics Textbook: Barnett/Ziegler/Byleen/Sobecki:
More informationSection 4.2 Polynomial Functions of Higher Degree
Section 4.2 Polynomial Functions of Higher Degree Polynomial Function P(x) P(x) = a degree 0 P(x) = ax +b (degree 1) Graph Horizontal line through (0,a) line with y intercept (0,b) and slope a P(x) = ax
More informationAlgebra 1.5 Year Long. Content Skills Learning Targets Assessment Resources & Technology CEQ: Chapters 1 2 Review
St. Michael Albertville High School Teacher: Mindi Sechser Algebra 1.5 Year Long August 2015 CEQ: Chapters 1 2 Review Variables, Function A. Simplification of Patterns, and Graphs Expressions LT1. I can
More informationUnit 2 Rational Functionals Exercises MHF 4UI Page 1
Unit 2 Rational Functionals Exercises MHF 4UI Page Exercises 2.: Division of Polynomials. Divide, assuming the divisor is not equal to zero. a) x 3 + 2x 2 7x + 4 ) x + ) b) 3x 4 4x 2 2x + 3 ) x 4) 7. *)
More informationa real number, a variable, or a product of a real number and one or more variables with whole number exponents a monomial or the sum of monomials
51 Polynomial Functions Objectives A2.A.APR.A.2 (formerly AAPR.A.3) Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function
More informationMission 1 Simplify and Multiply Rational Expressions
Algebra Honors Unit 6 Rational Functions Name Quest Review Questions Mission 1 Simplify and Multiply Rational Expressions 1) Compare the two functions represented below. Determine which of the following
More informationRadical Equations and Inequalities
16 LESSON Radical Equations and Inequalities Solving Radical Equations UNDERSTAND In a radical equation, there is a variable in the radicand. The radicand is the expression inside the radical symbol (
More informationCHAPTER 8A RATIONAL FUNCTIONS AND RADICAL FUNCTIONS Section Multiplying and Dividing Rational Expressions
Name Objectives: Period CHAPTER 8A RATIONAL FUNCTIONS AND RADICAL FUNCTIONS Section 8.3  Multiplying and Dividing Rational Expressions Multiply and divide rational expressions. Simplify rational expressions,
More informationUnit Lesson Topic CCSS
Advanced Algebra Tutorial NEW YORK This map correlates the individual topics of the Advanced Algebra Tutorial to specific Common Core State Standards. For more detailed information about these standard
More informationChapter 4 Polynomial and Rational Functions
Chapter Polynomial and Rational Functions  Polynomial Functions Pages 09 0 Check for Understanding. A zero is the value of the variable for which a polynomial function in one variable equals zero. A root
More informationADVANCED/HONORS ALGEBRA 2  SUMMER PACKET
NAME ADVANCED/HONORS ALGEBRA 2  SUMMER PACKET Part I. Order of Operations (PEMDAS) Parenthesis and other grouping symbols. Exponential expressions. Multiplication & Division. Addition & Subtraction. Tutorial:
More informationSection Properties of Rational Expressions
88 Section.  Properties of Rational Expressions Recall that a rational number is any number that can be written as the ratio of two integers where the integer in the denominator cannot be. Rational Numbers:
More informationAlgebra 1 Course Syllabus. Algebra 1, Part 1
Course Description: Algebra 1 Course Syllabus In Algebra 1, students will study the foundations of algebra, including the understanding of variables, expressions, and working with real numbers to simplify
More informationSOLUTIONS FOR PROBLEMS 130
. Answer: 5 Evaluate x x + 9 for x SOLUTIONS FOR PROBLEMS  0 When substituting x in x be sure to do the exponent before the multiplication by to get (). + 9 5 + When multiplying ( ) so that ( 7) ( ).
More information