© 2011 Pearson Education, Inc. Publishing as Prentice Hall. Remember that factoring out the GCF from the terms **of** a **polynomial** should always be the first step in factoring a **polynomial**. This will usually be followed by additional steps in the process. Factor **90** + 15y 2 – 18x – 3xy 2. **90** + 15y 2 – 18x – 3xy 2 = 3(30 + 5y 2 – 6x – xy 2 ) = 3(5 · 6 + 5/

have n distinct data points we can always fit a **polynomial** **of** order up to n-1. In the process what we claim to be random error is actually a systematic departure as the result **of** not fitting enough terms. Regression II; Bu-Ali Sina /1) Therefore, using the above F statistics we can test hypothesis H 0. Regression II; Bu-Ali Sina University Fall 2014 **90** Analysis **of** covariance Ch 8: Indicator Variables: Concurrent and coincide lines Regression II; Bu-Ali Sina University Fall 2014 91 In the concurrent /

division. A.y – 2 B.y – 1 C.y + 5 D.y – 1 Example 4 Divide **Polynomials** to Solve a Problem GEOMETRY The area **of** a rectangle is represented by 3x + **90**. Its length is (x – 3). Find (3x + **90**) ÷ (x – 3) to determine the width **of** the rectangle. Answer: So, represents the width **of** the rectangle. 99 (–) 3x – 9 3 Example 4 GEOMETRY The area/

– insert small items according to ratio v n / w n – clever analysis 15 Approximation Algorithms and Schemes Types **of** approximation algorithms. n Fully **polynomial**-time approximation scheme. n Constant factor. 16 Traveling Salesperson Problem TSP: Given a graph G = (V, E), /give instance where solution is almost factor **of** 2 from optimal. – m machines, m(m-1) jobs with **of** length 1, 1 job **of** length m – 10 machines, **90** jobs **of** length 1, 1 job **of** length 10 Machine 551525354555657585 Machine 661626364656667686 /

= x 2 + 19x + **90** 9x + 20 = 19x + **90** -70 = 10x x = -7 Application involving multiplication **of** **polynomials** A square painting is surrounded by a border 2 inches wide. If the area **of** the border is 96 square inches, find the dimensions **of** the painting. Application involving multiplication **of** **polynomials** 1.What am looking for? dimension **of** painting: x 2.What is known? area **of** border: 96 width **of** edge: 2 3/

. IF ANY TERMS ARE MISSING THENWRITE THE COEFFICIENT AS ZERO. 1.WRITE THE **POLYNOMIAL** IN COEFFICIENT FORM.IF THE DEGREE **OF** THE **POLYNOMIAL** IS 3 THEN THERE ARE 4 COEFFICIENTS. IF ANY TERMS ARE MISSING THENWRITE THE/ 2. EX. FIND THE REMAINDER WHEN THE **POLYNOMIAL** 2 X 4 – 6 X 3 + 2 X 2 – X + 2 IS DIVIDED BY X + 2. - 2 2 - 6 + 2 - 1 + 2 - 2 2 - 6 + 2 - 1 + 2 - 4 + 20 - 44 + **90** - 4 + 20 - 44 + **90** -------------------------------------------------- -------------------------------------------------- 2 - 10 + 22 - 45/

has (monotone) feasible interpolation if there is a (monotone) interpolant circuit for (A ∧ B) **of** size **polynomial** in the size **of** the shortest S-proof **of** (A ∧ B). Feasible interpolation property implies superpolynomial lower bounds (for S). Feasible Interpolation: Important /time SDPs! Systematic tightening: Lift-and-project hierarchies Intuition: Complicated convex body can be projection **of** simpler convex body Sherali-Adams ’**90** Lovász-Schrijver ’91 Lasserre ’01 Examples: n rounds integral hull time n O(/

**Polynomial** number **of** “simple” constraints + some **of** the combinatorial ones Add inequalities END YES NO **Polynomial** number **of** constraints for the separation problem as an LP **Polynomial** number **of** constraints to force that no violated combinatorial constraint exists LP Separation Paradigm **Polynomial** number **of** “simple” constraints **Polynomial** number **of** constraints for the separation problem as an LP **Polynomial** number **of**/: works only for small proteins (60 residues, **90** contacts) can be slow and involved: relies on/

- 8 4a - 8 = 0 4a = 8 a = 2 The **polynomial** becomes : = (x - 2)(x 2 + 4x - 45) x 3 + 2x 2 - 53x + **90** = (x - 3)(x + 5)(x - 9) x = 3, -5, 9 Quadratic Theory Higher Mathematics Solving Quadratic Equations There are three methods we can use to solve a quadratic equation **of** the form ax 2 + bx + c = 0 1.Factorising x/

R 2 that rotates a vector by **90** ◦ standard matrix **of** the **90** ◦ -rotation: No eigenvalues, no eigenvectors Characteristic **Polynomial** In general, a matrix A and RREF **of** A have different characteristic **polynomials**. Similar matrices have the same characteristic **polynomials** Different Eigenvalues The same Eigenvalues Characteristic **Polynomial** Question: What is the order **of** the characteristic **polynomial** **of** an n n matrix A? The characteristic **polynomial** **of** an n n matrix is indeed a/

· (x + 2) = (x + 2)(6 – y) Factoring Remember that factoring out the GCF from the terms **of** a **polynomial** should always be the first step in factoring a **polynomial**. This will usually be followed by additional steps in the process Example: Factor **90** + 15y 2 – 18x – 3xy 2. **90** + 15y 2 – 18x – 3xy 2 = 3(30 + 5y 2 – 6x – xy 2 ) = 3(5 · 6 + 5/

1: Example 2: Oblique Asymptotes (3) An oblique (or slant) asymptote exists only when the degree **of** numerator **polynomial** is exactly 1 higher than that **of** denominator **polynomial** (i.e., n = m + 1). How to find them? Divide the numerator **polynomial** by the denominator **polynomial** using long division, the quotient **polynomial** is the oblique asymptote. Example: Notes: A rational function can have more than one vertical asymptotes/

wt(Deg(P 1 )) ≤ k. October 18, 2011 Chicago: Testing Affine-Invariant Properties 35 **of** 37 Hopes Get a complete characterization **of** locally testable affine-invariant properties. Get a complete characterization **of** locally testable affine-invariant properties. Use codes **of** (**polynomially** large?) locality to build better LTCs/PCPs? Use codes **of** (**polynomially** large?) locality to build better LTCs/PCPs? In particular move from “domain = vector space” to/

wt(Deg(P 1 )) ≤ k. February 22, 2012 Invariance in Property Testing: Yale 37 **of** 39 Hopes Get a complete characterization **of** locally testable affine-invariant properties. Get a complete characterization **of** locally testable affine-invariant properties. Use codes **of** (**polynomially** large?) locality to build better LTCs/PCPs? Use codes **of** (**polynomially** large?) locality to build better LTCs/PCPs? In particular move from “domain = vector space” to/

laboratory. No! It appears almost always in a very broad class **of** random graphs [ Valiant, Roughgarden, EC ‘06 ] It has been long observed in many large cities, such as NY [ Kolata, New York Times ‘**90** ]. “It is just as likely to occur as not” [ /MinLatMod): with each edge e endowed with **polynomial** **of** degree d latency on flow x: modify the latency: So that: (i) the Euclidian distance **of** the coefficient’s vectors is minimum & (ii) the induced common latency gives the cost **of** optimum flow on G AGT 2012 SAMOS/

Classification Rates % Correct Classification No. **of** predictors: (9) (12) (13) (20) Comparison **of** Classification Rates % Correct Classification 96%92%89%**90**% No. **of** predictors: (9) (12) (13) (20) Comparison **of** Classification Rates % Correct Classification No. **of** predictors: (9) (12) (13) (20) Comparison **of** Classification Rates % Correct Classification No. **of** predictors: (9) (12) (13) (20) Comparison **of** Classification Rates Summary **of** findings Comparing **polynomial**-based tests & direct measurement-based tests/

to Multivariable Control Topics to be covered include: Multivariable Connections Multivariable Poles Multivariable Zeros Directions **of** Poles and Zeros Smith Form **of** a **Polynomial** Matrix Smith-McMillan Forms Matrix Fraction Description (MFD) Scaling Performance Specification Trade-offs in Frequency/with PM < **90**˚ (most practical systems) we have In conclusion ωB ( which is defined in terms **of** S ) and also ωc ( in terms **of** L ) are good indicators **of** closed-loop performance, while ωBT ( in terms **of** T ) may/

Reference on Numerical Algebraic Geometry up to 2005: A.J. Sommese and C.W. Wampler, Numerical solution **of** systems **of** **polynomials** arising in engineering and science, (2005), World Scientific Press. Recent articles are available at www.nd.edu/~/ space was beginning to be used, but the methods were a combination **of** differential topology and numerical analysis with homotopies tracked exclusively through real parameters. early **90**’s: algebraic geometric methods worked into the theory: great increase in security/

. AVOID quadratic terms [e.g. pH * pH (pH 2 ) (cf. multiple regression and **polynomial** terms)]. Can create an ARCH effect or warpage **of** ordination space. Try to avoid interaction terms except in clearly defined hypothesis-testing studies where the null hypothesis/ distributions, vegetation types, species richness, etc. To evaluate predictions, did 10-fold cross-validation, namely model with **90**% **of** the plots, predict with left-out 10%, and repeat 10 times. Compare predictions with actual observed data. Also/

–Verify whether indeed it is a solution –Both actions can be done in a “short” time (linear or low-degree **polynomial**), but the number **of** guesses, ND generated can grow often exponentially or more. Classical examples: –SAT –HC (Hamiltonian circuit problem in graph theory/p 2 ] = [0.8 0.2] : In the long term, the student passes 80% **of** the exams she attempts. Revisiting traditional models68 0,**90**,1 0,40,6 Adding input: Markov decision processes A probabilistic finite-state automaton with input is a probabilistic/

and correct single bit errors 0 0 5: DataLink Layer5-23 Cyclic Redundancy Check (CRC) r **Polynomial** code m Treat data bits as coefficients **of** n-bit **polynomial** m Choose r+1 bit generator **polynomial** G G well known – chosen in advance m Add r bits to packet so that message /CPU capacity at end-hosts m IP wins 5: DataLink Layer5-211 Frame Relay r Designed in late ‘80s, widely deployed in the ‘**90s** m Second-generation X.25 r Frame relay service: m no error control m no flow control m End-to-end congestion control m/

ax + b, where a and b are constants, is a linear function. 0 1 2 3 4 5 6 100 **90** 80 70 60 100 **90** 80 70 60 Scatter Plot A Linear Function f(x) = 2x + 80 Ex- 1, 2, 3, 4, 5/= ax +b and is an example **of** a **polynomial** function. However, **polynomial** functions **of** degree 2 or higher are nonlinear functions. To model nonlinear data we use **polynomial** functions **of** degree 2 or higher. Chapter 5 **Polynomial** Expressions and Functions 5.1 **Polynomial** Functions The following are examples **of** **polynomial** functions. f(x) = 3 Degree /

Factors: 1,2,3,4,6,9,12,18,36 Ex) 23 Prime. Factors: 1,23 Prime Factorization Ex) **90** = 2 ∙ 45 = 2 ∙ 3 ∙ 15 = 2 ∙ 3 ∙ 3 ∙ 5 OR use a factor tree: **90** 9 10 3 3 2 5 Prime Factorization **of** Negative Integers Ex) -140 = -1 ∙ 140 = -1 ∙ 2 ∙ 70 = -1 ∙ 2 ∙ 7/ · a 16a = 2 · 2 · 2 · 2 · a Factoring Using the (Reverse) Distributive Property First step is to find the prime factors **of** each term. Next step is to find the GCF **of** the terms in the **polynomial**. Ex) 12a²+ 16a 12a²= 2 · 2 · 3 · a · a 16a = 2 · 2 · 2 · 2 · a GCF =/

: 125x 3 + 27y 3 (5x) 3 + (3y) 3 a = 5x and b = 3y (5x + 3y) ((5x) 2 – (3y)(5x) + (3y) 2 ) (5x + 3y)(25x 2 – 15xy + 9y 2 ) Difference **of** Two Cubes a 3 - b 3 = (a - b)(a 2 + ab + b 2 ) Example: z 3 – 216 (z) 3 – (6) 3 a = z and b = 6 (z – 6) (z 2/ GCF = 3mn 3mn(m 3 + 64) 3mn(m 3 + 4 3 ) 3mn(m + 4)(m 2 – 4m +16) General Procedure for Factoring a **Polynomial** Factor by Grouping a = 9 b = -9 c = -10 a c = 9 -10 = -**90** Factor -**90** add -9 (6)(-15) 6 + -15 Sum in Two Cubes (a 3 + b 3 ) (a + b)(a – ab +b 2 ) a = m /

out the GCF Example: 2x 3 - 6x 2 = 2x 2 ( x – 3) Always make sure the remaining **polynomial**(s) are factored. Two Terms: Check for “DOTS” A 2 – C 2 (Difference **of** Two Squares) Example: x 2 – 4 = (x – 2 ) (x + 2) See if the binomials will/º angle = ½ hypotenuse Side opposite the 60º angle = ½ hypotenuse times √3 The larger leg equals the shorter leg times √3 30º- 60º- **90**º 45º- 45º- **90**º The ratio **of** the sides is 1 : 1 : √2 Side opposite the 45º angle = ½ hypotenuse times √2 Hypotenuse = s √2 where s = a leg/

velocities-11.136.31.49.0 Easting velocities-9.317.31,86.1 Teaching set residuals (mm/year)minmaxmeanStd. Northing velocities-4.13.**90**.01.2 Easting velocities-7.98.20.02.8 Table1 :The statistics **of** the differences between **polynomial** model and real velocities in the 62 teaching points. Testing set residuals (mm/year)minmaxmeanStd. Northing velocities-6.28.3-0.52/

the following situation: Node j = Node 1Node i = Node 2 δiδi δjδj Transformation with Matrix [T] We have : c = cos **90** o =0, and s = sin **90** o = 1.0, leading to: The above expression leads to: δ i = -12x10 -6 m and δ j = 0 or: We have /First Course in the FEM” 6 th Edition, Cengage Learning by Daryl Logan, 2017 Derive Interpolation Function We assume the traverse displacement **of** the beam element follows a linear **polynomial** function o the form: (4.41) in which a 1, a 2, a 3 and a 4 are constant coefficients x/

’**90**]) (Not obvious, [BlumLubyRubinfeld’**90**]) August 29-30, 2011 Rabin ’80: APT 4 **of** 22 July 29, 2011 Invariance in Property Testing: EPFL 5 History ( slightly abbreviated ) [Blum,Luby,Rubinfeld – S’**90**] [Blum,Luby,Rubinfeld – S’**90**]/**of** 22 Hopes Get a complete characterization **of** locally testable affine-invariant properties. Get a complete characterization **of** locally testable affine-invariant properties. Use codes **of** (**polynomially** large?) locality to build better LTCs/PCPs? Use codes **of** (**polynomially**/

Mathematics. A Wiley-Interscience publication. Chichester etc.: John Wiley & Sons. X, 465 p. (1985). MSC 2000: *00Bxx **90**-06Lawler, E.L.(ed.)Lenstra, J.K.(ed.)Rinnooy Kan, A.H.G.(ed.)Shmoys, D.B.(ed.)00Bxx90-06 / Martin Grötschel 81 Separation Algorithms Given a system **of** valid inequalities (possibly **of** exponential size). Is there a **polynomial** time algorithm (or a good heuristic) that, given a point, checks whether the point satisfies all inequalities **of** the system, and if not, finds an inequality /

Factors Chapter Summary 1.The process **of** rewriting a **polynomial** as a product **of** its factors is called factorization. 2.Factorization is the reverse process **of** expansion. 3.When each term **of** a **polynomial** has one or more common factors, we can factorize the **polynomial** by taking out common factors. P/)102 2 (100 2) 2 100 2 2(100)(2) 2 2 10 000 400 4 (b)89 2 (**90** 1) 2 **90** 2 2(**90**)(1) 1 2 8100 180 1 7921 10 404 4.4 Factorization by Taking out Common Factors Follow-up 4.12 (a)3a 2/

2.8 Third differences: 0.6 1 0.6 0.8 0.8Close Check It Out! Example 2 Continued Step 2 Determine the degree **of** the **polynomial**. Because the third differences are relatively close, a cubic function should be a good model. Step 3 Use the cubic regression feature on /**90**(5)3 + 2153.24(5)2 – 2183.29(5) + 3871.46 Based on the model, the opening value was about $11,479.76 in 1999. x 8 10 12 14 16 18 y 7.2 1.2 –8.3 –19.1 –29 –35.8 Lesson Quiz: Part I 1. Use finite differences to determine the degree **of** the **polynomial**/

2.8 Third differences: 0.6 1 0.6 0.8 0.8Close Check It Out! Example 2 Continued Step 2 Determine the degree **of** the **polynomial**. Because the third differences are relatively close, a cubic function should be a good model. Step 3 Use the cubic regression feature on /**90**(5)3 + 2153.24(5)2 – 2183.29(5) + 3871.46 Based on the model, the opening value was about $11,479.76 in 1999. x 8 10 12 14 16 18 y 7.2 1.2 –8.3 –19.1 –29 –35.8 Lesson Quiz: Part I 1. Use finite differences to determine the degree **of** the **polynomial**/

(0.073)- 1.950 (0.060)1.98 (0.105)2.53 (0.068) --0.03 (0.091) - 1.060 1.**90** R2R2 0.875 0.77 M 2 overfitting y =ß 1 x 1 + ß 2 x 2 + ß 3 x 3 + ε Standard /) – Regression splines – Smoothing splines Parametric (non-local influence) models – **Polynomials** – Non-linear curves – Fractional **polynomials** Intermediate between **polynomials** and non-linear curves 43 Fractional **polynomial** models Describe for one covariate, X – multiple regression later Fractional **polynomial** **of** degree m for X with powers p 1, …, p m is given /

mls) approximation Gaussian weight function locality idea: locally approximate surface with **polynomial** –compute reference plane –compute weighted least- squares fit **polynomial** implicit surface definition using a projection operator surface model moving least squares (/ for definition **of** approximating planes compute fundamental quadrics compute initial point-pair contraction candidates iterative simplification 2d example compute edge costs iterative simplification 2d example 60.02 20.03 140.04 5 **90**.09 10./

all are distinct. We have These also represent the zeros **of** the numerator **polynomial** Hence and which simplifies into From (19-**90**) and (19-92) we get (19-89) (19-**90**) (19-91) (19-92) which is at most **of** degree n – 1 in s2 vanishes at PILLAI i.e., the **polynomial** which is at most **of** degree n – 1 in s2 vanishes at (for n distinct/

Projective space was beginning to be used, but the methods were a combination **of** differential topology and numerical analysis with homotopies tracked exclusively through real parameters. early **90**’s: algebraic geometric methods worked into the theory: great increase in security,/ apart. Satellite Conference on Algebraic Geometry Segovia, Spain, August 16, 2006 67 Evaluation To 15 digits **of** accuracy one **of** the roots **of** this **polynomial** is a = 27.9999999999999. Evaluating p(a) to 15 digits, we find that p(a) /

f(r) is always less than kp(r) Exponential-time algorithm The time complexity function f(r) is not bounded by a **polynomial** Intractability and NP-completness Example (1) Suppose we need 10-5 seconds to solve an instance **of** a problem whose size is r=10 Then for an algorithm, whose time complexity function is linear in r we have, for/

ötschel 78 Separation Algorithms Given a system **of** valid inequalities (possibly **of** exponential size). Is there a **polynomial** time algorithm (or a good heuristic) that, given a point, checks whether the point satisfies all inequalities **of** the system, and if not, finds /: Separation ~ 3|V| variables ~1.5|V| constraints Column generation: Pricing. Martin Grötschel **90** A Pictorial History **of** Some TSP World Records Martin Grötschel 91 Some TSP World Records yearauthors# cities# variables 1954DFJ42/49820//

Normal, LogNormal Laguerre [0, ∞] Gamma Number **of** chaos **polynomials** used to represent output uncertainty depends on - Type **of** uncertainty in input- Distribution **of** input uncertainty - Number **of** terms in KLE **of** input - Degree **of** uncertainty propagation desired Materials Process Design and Control / 0.03 30 snapshots at equal intervals with ε 0 = 0.7; σ = 0.02 Using 5 out **of** a possible **90** POD basis vectors for the energy and momentum equations. 2D order 3 basis used for random dimension Basis info Other/

into for some applications. Such filters will have “linear phase” in the passband, but the intercept **of** such a filter at DC must be at +_ **90** degrees, and the filter must have a zero at DC. An asymmetric impulse response implies: –The/coefficient values. –How does this relate to filtering? If you convolve a set **of** factored **polynomials**, you get the original **polynomial** –That means that if you cascade sections with the **polynomials** implemented, you MULTIPLY the transfer functions. This is the same old duality in /

exist, it is necessary for the 52 resulting equations to be linearly dependent. This will occur if the determinant **of** M equals zero. It was not possible to expand the determinant symbolically. A numerical case was analyzed and a **polynomial** **of** degree 158 in the variable L 1 was obtained. A numerical example is presented next. 52 3-S: 1/ 4 + (L 1 4 – 440 L 1 2 -13824) L 2 2 – 8624 L 1 2 + 6773760 = 0 (3-33) (2.5 L 1 L 2 2 + **90** L 1 – 1600) L 3 3 + (-8.663 L 1 L 2 2 + 381.165 L 1 ) L 3 2 + [-2.5 L 1 L 2 4 + (-6 /

as Pearson Addison-Wesley Introduction to **Polynomials** 5.2 1.Identify monomials. 2.Identify the coefficient and degree **of** a monomial. 3.Classify **polynomials**. 4.Identify the degree **of** a **polynomial**. 5.Evaluate **polynomials**. 6.Write **polynomials** in descending order **of** degree. 7.Combine like terms. /each term in the **polynomial** by the monomial. Slide 5- 89 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Multiply. a. Solution a. 2p 6p 2 2p 2p –1 Slide 5- **90** Copyright © 2006 /

, ba} 1,3,6 2,10 4,13 5 7,15 8 9 11 12 14 b **90** Erice 2005, the Analysis **of** Patterns. Grammatical Inference **90** After determinizing, negative string aa is again accepted a b ab a X - ={aa, ab,/incremental. 12 6 Erice 2005, the Analysis **of** Patterns. Grammatical Inference 126 Properties (4) **Polynomial** aspects **Polynomial** characteristic sets **Polynomial** update time But not necessarily a **polynomial** number **of** mind changes 12 7 Erice 2005, the Analysis **of** Patterns. Grammatical Inference 127 Extensions Sakakibara built/

uncertain inputs can use combination **of** these **polynomials** for uncertainty representation Combinations **of** uncertain inputs can use combination **of** these **polynomials** for uncertainty representation Number **of** chaos **polynomials** used to represent output uncertainty depends on Number **of** chaos **polynomials** used to represent output uncertainty depends on - Type **of** uncertainty in input - Distribution **of** input uncertainty - Number **of** terms in KLE **of** input - Degree **of** uncertainty propagation desired (first order, second/

substitution constructs a new linear program with one less variable. Essentially, this is a contraction **of** k and l. Minimum Cost Flow - Strongly **Polynomial** Algorithms Introduction Minimum-Mean Cycle Canceling Algorithm Repeated Capacity Scaling Algorithm Enhanced Capacity Scaling Algorithm Summary What is this Contraction? **90** ● The contraction operation consists **of**: 1.Letting b(p) = b(k) + b(l) 2.Replacing each arc (i, k/

cutpoint 11 Inflation **of** type I errors (wrongly declaring a variable as important) Cutpoint selection in inner interval (here 10% - **90**%) **of** distribution **of** factor % significant Sample size Simulation study Type I error about 40% istead **of** 5% Increased /0, 0.5, 1, 2, 3} Power 0 means log X here ( conventional **polynomial** p 1 = 1, p 2 = 2,... ) Fractional **polynomial** models Describe for one covariate, X –multiple regression later Fractional **polynomial** **of** degree m for X with powers p 1, …, p m is given by FPm(X/

.2.3.2 Newton’s Divided Difference Interpolation 55 Example 3.9 Find interpolation **polynomial** that passes through (0,2), (1,3) dan (3,3) using Newton’s divided difference interpolation. Solution Table **of** divide difference for these three points is xy 02 1 13 0 33 56/.50.450.816686 60.60.550.738968 70.70.650.655406 80.80.750.569783 **90**.**90**.850.485537 1010.950.405555 TOTAL7.471308 67 3.3.3 Trapezoidal Rule A revision **of** the Rectangular Rule. While in the Rectangular Rule we make a rectangle for each subinterval/

3 -5x 2 + 10x- 23) (3x 3 – 5x 2 +x -11) + (-4x 3 -7x 2 +6x -23) (x 3 – 5x 2 + 4x) + (-4x 3 +6x -23) Multiplying **Polynomials** Use the distributive property and Laws **of** Exponents -5x 2 (3x 2 + 4x - 11) = -15x 4 - 20x 3 + 55x 2 (x+ 3)(x - 8)= x 2 - 8x = x 2 - 5x - 24 + 3x - 24 2x/. 0 1 2 3 4 5 6 7 8 HoursGrade 384 277 592 170 060 490 375 95 **90** 85 80 75 70 65 60 Education Write the Equation **of** the Line **of** Best Fit. 0 1 2 3 4 5 6 7 8 95 **90** 85 80 75 70 65 60 Rise = 30 Run = 4 Slope = 30 15 4 2 Intercept = 60 15/

**90**’s to present) – Colom, Ezpeleta & Tricas – Xie & Jeng – Zhou and his colleagues – Fanti & her colleagues – Roszkowska – Hsieh – Reveliotis, Lawley, Ferreira, Park and Choi A RAS taxonomy Structure **of** the process sequential logic Linear: each process is defined by a linear sequence **of** stages Disjunctive: A number **of**/ their acquisition Type 2: Unsafety Deadlock deadlock is **polynomially** identifiable. This kind **of** results are available for sub-classes **of** DIS-SU-RAS only. DC-RAS with “nested” resource /

describes the data. Third differences: 6.3 –2.3 0.5 42.20.**90**.1Not constant –1.8–1.3–0.8Not constant 0.5 Constant 1. Holt McDougal Algebra 2 Curve Fitting with **Polynomial** Models Use finite differences to determine the degree **of** the **polynomial** that best describes the data. Using Finite Differences to Determine Degree The x-values increase by a constant/

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