##### POLYNOMIAL FUNCTIONS A POLYNOMIAL is a monomial or a sum of monomials. A POLYNOMIAL IN ONE VARIABLE is a polynomial that contains only one variable. Example:

3 + 2i ) – ( 6 + 13i ) 3 + 2i – 6 – 13i –3 – 11i 4 Multiplying Complex Numbers Multiplying complex numbers is similar to multiplying polynomials and combining like terms. Perform the operation and write the result in standard form.( 6 – 2i )( 2 – 3i ) FOILFOIL 12 – 18i – 4i + 6i/ 2 12 – 22i + 6 ( -1 ) 6 – 22i We have seen that if a polynomial equation is of degree n, then counting multiple roots separately, the equation has n roots. This result is called the Fundamental Theorem of Algebra./

##### POLYNOMIAL PATTERNS Learning Task: 1. In the activation activity, we looked at four different polynomial functions. a. Let’s break down the word: poly-

+ 4x 4 – 7x 3 – 22x 2 +24x) 6. d. Sketch your own higher order e. Sketch your own higher order polynomial polynomial function (an equation function with symmetry about the origin. is not needed) with symmetry about the y-axis. f. Using these examples from /the turning points and the related intervals of increase and decrease, as you have determined previously for linear and quadratic polynomial functions. Record which turning points are relative minimum (the lowest point on a given portion of the graph) and/

##### 1 5.1 Polynomial Functions In this section, we will study the following topics: Identifying polynomial functions and their degree Determining end behavior.

or down by looking at the sign of “ a ” of the quadratic function, we can use the LEADING COEFFICIENT of higher degree polynomials to determine if the left and right ends of the graph rise or fall. We call this the END BEHAVIOR of the graph of the/ END BEHAVIOR of the graph (whether graph falls or rises as x approaches  and -  ). 3. Find the REAL ZEROS of the polynomial, if possible. These will be the x-intercepts of the graph. Watch out for repeated zeros. 4. Plot a few ADDITIONAL POINTS by substituting/

##### Holt Algebra 1 7-5 Polynomials A monomial is a number, a variable, or a product of numbers and variables with whole-number exponents. The degree of a monomial.

standard form is The leading coefficient is 1. x 5 + 9x 3 – 4x 2 + 16. Holt Algebra 1 7-5 Polynomials Some polynomials have special names based on their degree and the number of terms they have. Degree Name 0 1 2 Constant Linear Quadratic 3 4/ 8 + 6y 8 Identify like terms. Rearrange terms so that like terms are together. Combine like terms. Holt Algebra 1 7-5 Polynomials Polynomials can be added in either vertical or horizontal form. In vertical form, align the like terms and add: In horizontal form, use the/

##### Holt McDougal Algebra 2 3-1 Polynomials 3-1 Polynomials Holt Algebra 2 Warm Up Warm Up Lesson Presentation Lesson Presentation Lesson Quiz Lesson Quiz.

+ 3) + (–x 2 + x – 6) (–2x 2 – x 2 ) + (x) + (3 – 6) –3x 2 + x – 3 Holt McDougal Algebra 2 3-1 Polynomials Check It Out! Example 3a Add or subtract. Write your answer in standard form. (–36x 2 + 6x – 11) + (6x 2 + 16x 3 – 5) Add vertically. Write in standard form/ Align like terms. Add. –36x 2 + 6x – 11 +16x 3 + 6x 2 – 5 16x 3 – 30x 2 + 6x – 16 Holt McDougal Algebra 2 3-1 Polynomials Check It Out! Example 3b Add or subtract. Write your answer in standard form. (5x 3 + 12 + 6x 2 ) – (15x 2 + 3x – 2) Add the /

##### Chapter 4 Products and Factors of Polynomials. Section 4-1 Polynomials.

?  2x 2 Now factor:  2x 4 – 4x 3 + 8x 2  Factor out 2x 2  2x 2 (x 2 – 2x + 4) Perfect Square Trinomials  The polynomials in the form of a 2 + 2ab + b 2 and a 2 – 2ab + b 2 are the result of squaring a + b and a – b respectively Difference of Squares /-5} Zeros  A number r is a zero of a function f if f(r) = 0  You can find zeros using the same method that is used to solve polynomial equations Example  Find the zeros of f(x) = (x – 4) 3 – 4(3x – 16) 1: simplify 2: factor 3: set each factor = 0 Double/

##### 1 Real Numbers Polynomials Factoring Polynomials Rational Expressions Integral Exponents Solving Equations Rational Exponents and Radicals Quadratic Equations.

conclude that the correct factorization is Examples Use trial and error to factorize the following expressions: Factoring by Regrouping Sometimes a polynomial may be factored by regrouping and rearranging terms so that a common term can be factored out. Examples Factor the / Factor the first two terms Rearrange the terms Factor the common term x + 1 Factoring by Regrouping Sometimes a polynomial may be factored by regrouping and rearranging terms so that a common term can be factored out. Examples Factor the/

##### TOPICS COVERED  Polynomials in One Variable  Zeroes of a polynomial  Remainder Theorem  Factor Theorem  Algebraic Identities.

and –2 is the coefficient of x 0. (Remember, x 0 = 1)  The coefficient of x in x 2 – x + 7 is –1. Examples of Polynomials in one variable  Algebraic expressions like 2x, x 2 + 2x, x 3 – x 2 + 4x + 7 have only whole numbers as the exponents of the variable/can also be checked without applying the Factor Theorem, since 2x + 4 = 2(x + 2). Practice Questions 1.Determine whether x+1 is a factor of the following polynomials, x ³ -x ² -(2+  2)x+  2 2. Find the value of k,if x-1 is a factor of p(x) in each of /

##### Holt McDougal Algebra 1 6-5 Multiplying Polynomials 6-5 Multiplying Polynomials Holt Algebra 1 Warm Up Warm Up Lesson Presentation Lesson Presentation.

4b 4 2a 2 + 7ab 2 – 4b 4 Use the FOIL method. Multiply. Combine like terms. Holt McDougal Algebra 1 6-5 Multiplying Polynomials To multiply polynomials with more than two terms, you can use the Distributive Property several times. Multiply (5x + 3) by (2x 2 + 10x – 6): / 11x 3 – 4x 2 – 21x – 7 Combine like terms. Holt McDougal Algebra 1 6-5 Multiplying Polynomials A polynomial with m terms multiplied by a polynomial with n terms has a product that, before simplifying has mn terms. In Example 4A, there are 2/

##### October,2006 Higher- Degree Polynomials Peter L. Montgomery Microsoft Research and CWI 1 Abstract The Number Field Sieve is asymptotically the fastest.

no small prime factors. Also best known algorithm for discrete logarithms modulo large primes. October,2006 Higher- Degree Polynomials Peter L. Montgomery Microsoft Research and CWI 4 SNFS and GNFS Special Number Field Sieve (SNFS) –Number/ M = 37570227807001155896638712233675454511 P = 12722245648421103686881 = 11. 31. 61. 71. 191. 331. 461. 521. 691. 821 October,2006 Higher- Degree Polynomials Peter L. Montgomery Microsoft Research and CWI 34 Norm Sizes for RSA200 a 5 ≈ 2 3. 3 5. 5. 7. 13. 422861 /

##### Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Chapter 10 Exponents and Polynomials.

Helpful Hint When multiplying two binomials, you may always use the FOIL order or method. When multiplying any two polynomials, you may always use the distributive property to find the product. Martin-Gay, Prealgebra & Introductory Algebra, 3ed 63/until you can’t bring down or divide any longer. We will incorporate this same repeated technique with dividing polynomials. Dividing Polynomials Martin-Gay, Prealgebra & Introductory Algebra, 3ed 70 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice /

##### Holt Algebra 1 7-7 Multiplying Polynomials 7-7 Multiplying Polynomials Holt Algebra 1 Warm Up Warm Up Lesson Presentation Lesson Presentation Lesson Quiz.

together. Multiply. (6 3)(y 3 y 5 )   (3 9)(m m 2 )(n 2  n)   Holt Algebra 1 7-7 Multiplying Polynomials Multiply. Example 1C: Multiplying Monomials Group factors with like bases together. Multiply.       222 1 12 4 tstt s s     gg/ 4b 4 2a 2 + 7ab 2 – 4b 4 Use the FOIL method. Multiply. Combine like terms. Holt Algebra 1 7-7 Multiplying Polynomials To multiply polynomials with more than two terms, you can use the Distributive Property several times. Multiply (5x + 3) by (2x 2 + 10x – /

##### Pre-Algebra 13-1 Polynomials Pre-Algebra HOMEWORK Page 654 #1-14.

numbers. Monomials2n, x 3, 4a 4 b 3, 7 Not monomialsp 2.4, 2 x, √x, g2g2 5 Pre-Algebra 13-1 Polynomials monomialnot a monomial 3 and 4 are whole numbers. Additional Example 1: Identifying Monomials Determine whether each expression is a monomial. y does not/, 250 for v, and 20 for s. Simplify. 170 The rocket is 170 ft high 15 seconds after launching. Pre-Algebra 13-1 Polynomials Lesson Quiz noyes Insert Lesson Title Here trinomialbinomial 5 3 Determine whether each expression is a monomial. 1. 5a 2 z 4 2. 3√/

##### 7-6 Polynomials Warm Up Warm Up Lesson Presentation Lesson Presentation California Standards California StandardsPreview.

a single variable. The degree of a monomial is the sum of the exponents of the variables. A constant has degree 0. 7-6 Polynomials Additional Example 1: Finding the Degree of a Monomial Find the degree of each monomial. A. 4p4q34p4q3 The degree is 7. Add the exponents/ after 3 seconds? Additional Example 5 Continued After 3 seconds the lip balm will be 76 feet above the water. 7-6 Polynomials Check It Out! Example 4 What if…? Another firework with a 5-second fuse is launched from the same platform at a speed/

##### Holt Algebra 1 7-5 Polynomials 7-5 Polynomials Holt Algebra 1 Warm Up Warm Up Lesson Presentation Lesson Presentation Lesson Quiz Lesson Quiz.

5 4 713 2 69 4 1 2 –1 Holt Algebra 1 7-5 Polynomials Classify polynomials and write polynomials in standard form. Evaluate polynomial expressions. Objectives Holt Algebra 1 7-5 Polynomials A _______ is a number, a variable, or a product of numbers and/–7x 5 + 4x 2 + 6x + 9. The standard form is The leading coefficient is –7. Holt Algebra 1 7-5 Polynomials Write the polynomial in standard form. Then give the leading coefficient. Example 3B Find the degree of each term. Then arrange them in descending order: y/

respectively. Because 25 = 25x 0, 25 is referred to as a constant. 8 Determine whether an expression is a polynomial Polynomials A polynomial is an algebraic expression that is a single term or the sum of several terms containing whole-number exponents on the variables. /8, the degree of the last term 9, the degree of the first term 19 Find the degree of a polynomial If the polynomial contains a single variable, we usually write it with its exponents in descending order where the term with the highest degree/

##### Polynomial Equivalent Layer Valéria C. F. Barbosa* Vanderlei C. Oliveira Jr Observatório Nacional.

       Q B00 0B0 00B Β     2 1 Q equivalent-source windows Estimated polynomial coefficients How does the Polynomial Equivalent Layer work? Polynomial Equivalent Layer Step 1: N E Potential-field observations Depth Equivalent layer with Q equivalent-source windows c* Physical-property distribution Computed/ solves a 2,500 × 2,500 system Real Test Computed magnetization-intensity distribution obtained by Polynomial Equivalent Layer (PEL) N Real Test N Observed (black lines and grayscale map) and/

##### Precalculus Polynomial & Rational --- Part One V. J. Motto.

Rational Zeros (Leading Coefficient 1) The rational zeros of P are 1 and –2. E.g. 2—Using the Theorem to Factor a Polynomial Factor the polynomial P(x) = 2x 3 + x 2 – 13x + 6 By the theorem, the rational zeros of P are of the form/around 1637. Variation in Sign To describe this rule, we need the concept of variation in sign. Suppose P(x) is a polynomial with real coefficients, written with descending powers of x (and omitting powers with coefficient 0). A variation in sign occurs whenever adjacent /

##### 10-6 Dividing Polynomials Warm Up Warm Up Lesson Presentation Lesson Presentation California Standards California StandardsPreview.

2-5 as necessary until you get 0 or until the degree of the remainder is less than the degree of the binomial. 10-6 Dividing Polynomials Additional Example 3A: Polynomial Long Division Divide using long division. Check your answer. (x 2 +10x + 21) ÷ (x + 3) x 2 + 10x /term of the divisor to get the first term of the quotient. x 2 + 10x + 21 ) Step 2x + 3 x 10-6 Dividing Polynomials Additional Example 3A Continued Divide using long division. (x 2 +10x + 21) ÷ (x + 3) Multiply the first term of the quotient by/

##### Polynomials Jordi Cortadella Department of Computer Science.

// returned through the parameters Q and R. void PolyDivision(const Polynomial& A, const Polynomial& B, Polynomial& Q, Polynomial& R); Introduction to Programming© Dept. CS, UPC13 ABRQ Polynomial long division Introduction to Programming© Dept. CS, UPC14 26-310201/// Index to the next leading coef. of Q } PolyNormalize(R); } Introduction to Programming© Dept. CS, UPC15 GCD of two polynomials Introduction to Programming© Dept. CS, UPC16 Example: Re-visiting Euclidean algorithm for gcd // gcd(a, 0) = a // gcd(/

##### Section Concepts 2.1 Addition and Subtraction of Polynomials Slide 1 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction.

McGraw-Hill Companies, Inc. Permission required for reproduction or display. One commonly used algebraic expression is called a polynomial. A polynomial in one variable, x, is defined as a single term or a sum of terms of the form/the same variables, and the corresponding variables are raised to the same powers. Section 2.1 Addition and Subtraction of Polynomials 2.Addition of Polynomials Slide 11 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Recall that the /

##### Essential Question: How can you solve a higher-degree (larger than 2) polynomial equation? Students will write a summary describing the methods used to.

Question: How can you solve a higher-degree (larger than 2) polynomial equation? Students will write a summary describing the methods used to solve higher degree polynomials. Find the real-number solutions to the equation. Example 16 Essential / Essential Question: How can you solve a higher-degree (larger than 2) polynomial equation? Students will write a summary describing the methods used to solve higher degree polynomials. Factoring Steps 1) Greatest Common Factor 2) Two terms …Difference of Squares/

##### UBC March 20071 The Evergreen Project: The Promise of Polynomials to Boost CSP/SAT Techniques* Karl J. Lieberherr Northeastern University Boston joint.

pathways? UBC March 200792 Conclusions Presented SPOT, a family of MAX-CSP solvers based on look-ahead polynomials and non-chronological backtracking. SPOT has a desirable property: P-optimal. SPOT can be implemented very efficiently/ of clause learning: other clause learning schemes learn clauses implied from superresolvents by UnitPropagation. Resolution and Superresolution are polynomially equivalent (1977, Beame et al. (2004)). UBC March 2007124 Superresolution Mother of clause learning: minimal elements/

##### Holt McDougal Algebra 1 7-8 Multiplying Polynomials 7-8 Multiplying Polynomials Holt Algebra 1 Warm Up Warm Up Lesson Presentation Lesson Presentation.

2 a 2 + 7 ab 2 – 4 b 4 Use the FOIL method. Multiply. Combine like terms. Holt McDougal Algebra 1 7-8 Multiplying Polynomials To multiply polynomials with more than two terms, you can use the Distributive Property several times. Multiply (5 x + 3) by (2 x 2 + 10 x/ x 3 – 4 x 2 – 21 x – 7 Combine like terms. Holt McDougal Algebra 1 7-8 Multiplying Polynomials A polynomial with m terms multiplied by a polynomial with n terms has a product that, before simplifying has mn terms. In Example 4A, there are 2 3, or/

##### 7-8 Multiplying Polynomials Warm Up Warm Up Lesson Presentation Lesson Presentation California Standards California StandardsPreview.

+ 8ab 2 – ab 2 – 4b 4 2a 2 + 7ab 2 – 4b 4 Use the FOIL method. Multiply. Combine like terms. 7-8 Multiplying Polynomials To multiply polynomials with more than two terms, you can use the Distributive Property several times. Multiply (5x + 3) by (2x 2 + 10x – 6): (5x + / the rectangle. 3x 4 + 13x 3 + 4x 2 – 21x – 7 Combine like terms. 7-8 Multiplying Polynomials A polynomial with m terms multiplied by a polynomial with n terms has a product that, before simplifying has mn terms. In Example 4A, there are 2 3,/

##### Review 10.1- 10.4 Polynomials. Monomials - a number, a variable, or a product of a number and one or more variables. 4x, 20x 2 yw 3, -3, a 2 b 3, and.

+ 4 = (3x 2 – 2 x 2 ) + (– 5x + x) + (3 + 4) Multiplying Polynomials Distribute and FOIL Polynomials * Polynomials Multiplying a Polynomial by another Polynomial requires more than one distributing step. Multiply: (2a + 7b)(3a + 5b) Distribute 2a(3a + 5b) and distribute/5b) When multiplying by 3a, line up the first term under 3a. + Add like terms: 6a 2 + 31ab + 35b 2 Polynomials * Polynomials Multiply the following polynomials: Polynomials * Polynomials (x + 5) x (2x + -1) -x + -5 2x 2 + 10x + 2x 2 + 9x + -/

##### Holt Algebra 1 7-7 Multiplying Polynomials 7-7 Multiplying Polynomials Holt Algebra 1 Warm Up Warm Up Lesson Presentation Lesson Presentation Lesson Quiz.

2 – 4b 4 2a 2 + 7ab 2 – 4b 4 Use the FOIL method. Multiply. Combine like terms. Holt Algebra 1 7-7 Multiplying Polynomials To multiply polynomials with more than two terms, you can use the Distributive Property several times. Multiply (5x + 3) by (2x 2 + 10x – 6): (/ 3x 4 + 13x 3 + 4x 2 – 21x – 7 Combine like terms. Holt Algebra 1 7-7 Multiplying Polynomials A polynomial with m terms multiplied by a polynomial with n terms has a product that, before simplifying has mn terms. In Example 4A, there are 2 3, or/

##### Polynomials. Polynomial comes from poly- (meaning "many") and -nomial (in this case meaning "term")... so it says "many terms“

. For example multiply 2x (3x + 1) 3x+1 2x 6x 2 + 2x Multiplying Polynomials Distribute and FOIL Polynomials * Polynomials Multiplying a Polynomial by another Polynomial requires more than one distributing step. Multiply: (2a + 7b)(3a + 5b) Distribute 2a/x (3a + 5b) When multiplying by 3a, line up the first term under 3a. + Add like terms: 6a 2 + 31ab + 35b 2 Polynomials * Polynomials Multiply the following polynomials: Polynomials * Polynomials (x + 5) x (2x + -1) -x + -5 2x 2 + 10x + 2x 2 + 9x + -5 (3w + -/

##### Holt McDougal Algebra 2 6-1 Polynomials 6-1 Polynomials Holt Algebra 2 Warm Up Warm Up Lesson Presentation Lesson Presentation Lesson Quiz Lesson Quiz.

0.75 – m The degree of a monomial is the sum of the exponents of the variables. 1 2 Holt McDougal Algebra 2 6-1 Polynomials Identify the degree of each monomial. Example 1: Identifying the Degree of a Monomial A. z 6 B. 5.6 The degree is 6./ + 6x – 11) + (6x 2 + 16x 3 – 5) D. (5x 3 + 12 + 6x 2 ) – (15x 2 + 3x – 2) Holt McDougal Algebra 2 6-1 Polynomials Example 4: Work Application The cost of manufacturing a certain product can be approximated by f(x) = 3x 3 – 18x + 45, where x is the number of units of the/

##### Holt McDougal Algebra 2 Polynomials Identify, evaluate, add, and subtract polynomials. Classify and graph polynomials. Objectives.

2x 2 + 3) + (–x 2 + x – 6) (–2x 2 – x 2 ) + (x) + (3 – 6) –3x 2 + x – 3 Holt McDougal Algebra 2 Polynomials Check It Out! Example 3a Add or subtract. Write your answer in standard form. (–36x 2 + 6x – 11) + (6x 2 + 16x 3 – 5) Add vertically. Write in standard/ form. Align like terms. Add. –36x 2 + 6x – 11 +16x 3 + 6x 2 – 5 16x 3 – 30x 2 + 6x – 16 Holt McDougal Algebra 2 Polynomials Check It Out! Example 3b Add or subtract. Write your answer in standard form. (5x 3 + 12 + 6x 2 ) – (15x 2 + 3x – 2) Add the/

##### UBC March 20071 The Evergreen Project: The Promise of Polynomials to Boost CSP/SAT Techniques* Karl J. Lieberherr Northeastern University Boston joint.

feature of our algorithm (SPOT) is that it maintains two different formulations: the CSP formulation and the polynomial formulation. UBC March 20077 Gomes/Shmoys The hybrid nature of their algorithm results from the combination of / of clause learning: other clause learning schemes learn clauses implied from superresolvents by UnitPropagation. Resolution and Superresolution are polynomially equivalent (1977, Beame et al. (2004)). UBC March 200769 Superresolution Mother of clause learning: minimal elements /

##### Chapter 5 Exponents, Polynomials, and Polynomial Functions.

– x (6 + y 2 )) = 3(6 + y 2 )(5 – x) Factoring Example: Martin-Gay, Intermediate Algebra, 5ed 13 Factoring polynomials often involves additional techniques after initially factoring out the GCF. One technique is factoring by grouping. Factor xy + y + 2x + 2 by grouping. /Oops, this is the sum of squares, not the difference of squares, so it can’t be factored. This polynomial is a prime polynomial. Difference of Two Squares Example: Martin-Gay, Intermediate Algebra, 5ed 50 Factor 36x 2 – 64. Remember that/

##### Holt McDougal Algebra 2 Investigating Graphs of Polynomial Functions Investigating Graphs of Polynomial Functions Holt Algebra 2 Warm Up Warm Up Lesson.

x +∞, P(x) +∞. P(x) is of even degree with a positive leading coefficient. Holt McDougal Algebra 2 Investigating Graphs of Polynomial Functions Check It Out! Example 2a Identify whether the function graphed has an odd or even degree and a positive or negative leading /to increasing. A turning point corresponds to a local maximum or minimum. Holt McDougal Algebra 2 Investigating Graphs of Polynomial Functions A polynomial function of degree n has at most n – 1 turning points and at most n x-intercepts. If /

##### Holt Algebra 1 12-6 Dividing Polynomials 12-6 Dividing Polynomials Holt Algebra 1 Warm Up Warm Up Lesson Presentation Lesson Presentation Lesson Quiz Lesson.

x 2 + 2x x + 2 0 (x 2 + 3x + 2) ÷ (x + 2) Divisor Quotient Dividend Holt Algebra 1 12-6 Dividing Polynomials Holt Algebra 1 12-6 Dividing Polynomials Example 3A: Polynomial Long Division Divide using long division. (x 2 +10x + 21) ÷ (x + 3) x 2 + 10x + 21 ) Step 1x + 3/ divisor to get the first term of the quotient. x 2 + 10x + 21 ) Step 2x + 3 x Holt Algebra 1 12-6 Dividing Polynomials Example 3A Continued Divide using long division. (x 2 +10x + 21) ÷ (x + 3) Multiply the first term of the quotient by the/

##### Polynomials, Curve Fitting and Interpolation. In this chapter will study Polynomials – functions of a special form that arise often in science and engineering.

cant model y = 0 INTERPOLATION Interpolation is estimating values between data points. MATLAB can do interpolation with polynomials or the Fourier transform Wont discuss Fourier-transform interpolation in this book One-dimensional interpolation: linear interpolation is/ changes at every data point Can get smoother interpolations by using quadratic or cubic splines, which are polynomials whose coefficients are based only on data points near interpolated point INTERPOLATION MATLAB function interp1() does one-/

##### UNIT 2, LESSON 5 DESCARTES LAW OF SIGNS. FINDING POLYNOMIALS WITH GIVEN ZEROS If we are given the zeros of a polynomial, we can generate the polynomial.

Factor into linear factors. RATIONAL ZEROS THEOREM Factor into linear factors. DESCARTES RULE OF SIGNS The number of positive roots of a polynomial with real coefficients is equal to the number of "changes of sign" in the list of coefficients, or is less than/ there are at most 2 positive roots (maybe less). HOW MANY OF THE ROOTS ARE NEGATIVE? http://www.mathsisfun.com/algebra/polynomials-rule-signs.html By doing a similar calculation we can find out how many roots are negative...... but first we need to put/

##### Holt McDougal Algebra 2 3-1 Polynomials 3-1 Polynomials Holt Algebra 2 Warm Up Warm Up Lesson Presentation Lesson Presentation Lesson Quiz Lesson Quiz.

0.75 – m The degree of a monomial is the sum of the exponents of the variables. 1 2 Holt McDougal Algebra 2 3-1 Polynomials Identify the degree of each monomial. Example 1: Identifying the Degree of a Monomial A. z 6 Identify the exponent. B. 5.6 /– 18(0) + 45 = 45 f(200) = 3(200) 3 – 18(200) + 45 = 23,996,445 Holt McDougal Algebra 2 3-1 Polynomials Check It Out! Example 4 Cardiac output is the amount of blood pumped through the heart. The output is measured by a technique called dye dilution. For a patient/

##### POLYNOMIALS – Product of Polynomials Binomial times a binomial METHOD 1 : F.O.I.L. FOIL is an acronym for First, Outside, Inside, Last when multiplying.

when multiplying. EXAMPLE : Multiply First – the first terms in each binomial Outside – the outside terms of each binomial POLYNOMIALS – Product of Polynomials Binomial times a binomial METHOD 1 : F.O.I.L. FOIL is an acronym for First, Outside, Inside, Last/the first terms in each binomial Outside – the outside terms of each binomial Inside – the inside terms of each binomial POLYNOMIALS – Product of Polynomials Binomial times a binomial METHOD 1 : F.O.I.L. FOIL is an acronym for First, Outside, Inside, Last/

##### Roots & Zeros of Polynomials I How the roots, solutions, zeros, x-intercepts and factors of a polynomial function are related.

Division The Remainder Theorem Remainder = 1 The Factor Theorem Since the remainder is 0, x-3 is a factor of the polynomial. When you divide the polynomial by one of the binomial factors, the quotient is called a depressed equation. The Factor Theorem (x-2) Is NOT /or root at x = -1, the other 2 roots must have imaginary components. Descartes’ Rule of Signs Arrange the terms of the polynomial P(x) in descending degree: The number of times the coefficients of the terms of P(x) change sign = the number of /

coefficient of the first term is called the leading coefficient. Example: 3x 4 + 5x 2 – 7x + 1 and 3 is the leading coefficient. Vocab Write the polynomial in standard form. Then give the leading coefficient. 1. 6x – 7x 5 + 4x 2 + 9 2. 16 – 4x 2 + x 5 + 9x 3/ like terms. d. 9b 3 c 2 + -4b 3 + 5c 2 + 5b 3 c 2 – 13b 3 c 2 Let’s Practice… Polynomials can be added in either vertical or horizontal form. In vertical form, align the like terms and add: In horizontal form, use the Associative and Commutative /

##### Maclaurin and Taylor Polynomials Objective: Improve on the local linear approximation for higher order polynomials.

x = 1; that is, on this interval. Thus, we can take M = e to obtain Example 6 Use an nth Maclaurin polynomial for to approximate e to five decimal places. Unfortunately, this inequality is not very useful because it involves e, which is the very quantity/ < 3, then we can use this value. Although less precise, it is more easily applied. Example 6 Use an nth Maclaurin polynomial for to approximate e to five decimal places. Thus, we can achieve five decimal-place accuracy by choosing n so that or This happens/

##### Holt McDougal Algebra 1 6-5 Multiplying Polynomials Warm Up Evaluate. 1. 3 2 3. 10 2 Simplify. 4. 2 3  2 4 6. (5 3 ) 2 9 16 100 2727 2. 242. 24 5. y 5.

4b 4 2a 2 + 7ab 2 – 4b 4 Use the FOIL method. Multiply. Combine like terms. Holt McDougal Algebra 1 6-5 Multiplying Polynomials To multiply polynomials with more than two terms, you can use the Distributive Property several times. Multiply (5x + 3) by (2x 2 + 10x – 6): / 11x 3 – 4x 2 – 21x – 7 Combine like terms. Holt McDougal Algebra 1 6-5 Multiplying Polynomials A polynomial with m terms multiplied by a polynomial with n terms has a product that, before simplifying has mn terms. In Example 4A, there are 2/

##### 14-3 Adding Polynomials Warm Up Combine like terms. 1. 9x + 4x2. –3y + 7y 3. 7n + (–8n) + 12n Find the perimeter of each rectangle. 4. a 10 ft by 12 ft.

3(2x 2 – x) + x 2 + 1 13x 4y4y 11n 44 ft 26 m 7x 2 – 3x + 1 14-3 Adding Polynomials Learn to add polynomials. 14-3 Adding Polynomials Example 1A: Adding Polynomials Horizontally Add. (5x 3 + x 2 + 2) + (4x 3 + 6x 2 ) (5x + x + 2) + (4x/ 2 + b + 1) = b 2 + 4b – 1 – 7b 2 + b + 1 = – 6b 2 + 5b Add the opposite. Associative property. Combine like terms. 14-3 Adding Polynomials Example 2A Subtract. (2y 3 + 3y + 5) – (4y 3 + 3y + 5) Add the opposite. Associative property. = – 2y 3 Combine like terms. = (2y 3 + 3y /

##### Holt McDougal Algebra 1 6-3 Polynomials Warm Up Evaluate each expression for the given value of x. 1. 2x + 3; x = 22. x 2 + 4; x = –3 3. –4x – 2; x = –14.

713 2 69 4 1 2 –5 Holt McDougal Algebra 1 6-3 Polynomials Classify polynomials and write polynomials in standard form. Evaluate polynomial expressions. Objectives Holt McDougal Algebra 1 6-3 Polynomials A monomial is a number, a variable, or a product of numbers and/standard form is The leading coefficient is –3. –3y 8 + 18y 5 + 14y. Holt McDougal Algebra 1 6-3 Polynomials Some polynomials have special names based on their degree and the number of terms they have. Degree Name 0 1 2 Constant Linear Quadratic/

##### 7-8 Multiplying Polynomials To multiply monomials and polynomials, you will use some of the properties of exponents that you learned earlier in this chapter.

Group factors with like bases together. Multiply. (6 3)(y 3 y 5 )   (3 9)(m m 2 )(n 2  n)   7-8 Multiplying Polynomials Multiply. Additional Example 1C: Multiplying Monomials Group factors with like bases together. Multiply.       222 1 12 4 tstt s s     / rectangle. 3x 4 + 13x 3 + 4x 2 – 21x – 7 Combine like terms. 7-8 Multiplying Polynomials A polynomial with m terms multiplied by a polynomial with n terms has a product that, before simplifying has mn terms. In Example 4A, there are 2 3/

##### Three background concepts: 1.Decision Problems: output yes/no 2. Nondeterministic algorithm: certificate additional input 3. Polynomial transformation:

bound for Comparison Sort? Problems may be classified by such bounds An important, but “unknown” classification would be: –Polynomial-class versus “Exponential-class” Unfortunately: Life is much more complicated! –There is no E-class of problems (with/ expressed Problem instance: Specific values for Input, –Remember: Output is T/F in decision problems (C) Debasis Mitra POLYNOMIAL PROBLEM-TRANSFORMATION Problem X can be transformed to another problem Y Problem Transformation: an algorithm –T XY (input of X/

##### Slide 2- 1. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Chapter 2 Polynomial, Power, and Rational Functions.

2- 4 Quick Review Slide 2- 5 Quick Review Solutions Slide 2- 6 What you’ll learn about Polynomial Functions Linear Functions and Their Graphs Average Rate of Change Linear Correlation and Modeling Quadratic Functions and Their Graphs Applications/Slide 2- 46 Quick Review Solutions Slide 2- 47 What you’ll learn about Graphs of Polynomial Functions End Behavior of Polynomial Functions Zeros of Polynomial Functions Intermediate Value Theorem Modeling … and why These topics are important in modeling and can be/

##### Real Zeros of Polynomial Functions Section 2.3. Objectives Use long division to divide polynomials by other polynomials. Use synthetic division to divide.

t bring down or divide any longer. We will incorporate this same repeated technique with dividing polynomials. Dividing Polynomials Division of Polynomials Dividing Polynomials Long division of polynomials is similar to long division of whole numbers. dividend = (quotient divisor) + remainder /see this as f(x) = [q(x) ∙ (x – a)] + f(a). 2.The quotient is always a polynomial with one degree less than f(x). –Synthetic division is helpful in solving these problems (this can also be called synthetic substitution/

##### Copyright © 2013, 2009, 2005 Pearson Education, Inc. 1 3 Polynomial and Rational Functions Copyright © 2013, 2009, 2005 Pearson Education, Inc.

0.26794919, 2.4142136, and 3.7320508. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 62 Example 8 EXAMINING A POLYNOMIAL MODEL The table shows the number of transactions, in millions, by users of bank debit cards for selected years. Copyright © 2013/The best-fitting cubic function is shown and is Copyright © 2013, 2009, 2005 Pearson Education, Inc. 66 Example 8 EXAMINING A POLYNOMIAL MODEL (c) Repeat part (a) for a quartic function (degree 4). The best-fitting quartic function is shown and is Solution/