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./

+ 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/

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/

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/

+ 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 /

? 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/

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/

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 /

- 7x² + 5x - 7 Home ADDING/SUBTRACTING **POLYNOMIALS** 300 Add the given **polynomials**: Answer ADDING/SUBTRACTING **POLYNOMIALS** 300 What is: Home ADDING/SUBTRACTING **POLYNOMIALS** 400 Subtract the **polynomials**: Answer ADDING/SUBTRACTING **POLYNOMIALS** 400 What is: Home ADDING/SUBTRACTING **POLYNOMIALS** 500 Add and subtract the **polynomials**: Answer ADDING/SUBTRACTING **POLYNOMIALS** 500 What is: Home MULTIPLYING/ DIVIDING **POLYNOMIALS** 100 Multiply: (3x + 3)(4x – 2) Answer MULTIPLYING/ DIVIDING **POLYNOMIALS** 100 What is: 12x² + 6x - 6/

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/

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 /

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 /

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 – /

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√/

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/

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/

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/

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 /

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/

// 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(/

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 /

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/

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/

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/

+ 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,/

+ 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 + -/

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/

. 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 + -/

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/

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/

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 /

– 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/

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 /

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/

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-/

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/

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/

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/

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 /

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/

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/

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 /

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/

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/

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/

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/

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/

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/

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