8-5 Dividing Polynomials
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NP-Hardness of Deciding Convexity of Quartic Polynomials and Related Problems
NP-hardness of Deciding Convexity of Quartic Polynomials and Related Problems Amir Ali Ahmadi, Alex Olshevsky, Pablo A. Parrilo, and John N. Tsitsiklis ∗y Abstract We show that unless P=NP, there exists no polynomial time (or even pseudo-polynomial time) algorithm that can decide whether a multivariate polynomial of degree four (or higher even degree) is globally convex. This solves a problem that has been open since 1992 when N. Z. Shor asked for the complexity of deciding convexity for quartic polynomials. We also prove that deciding strict convexity, strong convexity, quasiconvexity, and pseudoconvexity of polynomials of even degree four or higher is strongly NP-hard. By contrast, we show that quasiconvexity and pseudoconvexity of odd degree polynomials can be decided in polynomial time. 1 Introduction The role of convexity in modern day mathematical programming has proven to be remarkably fundamental, to the point that tractability of an optimization problem is nowadays assessed, more often than not, by whether or not the problem benefits from some sort of underlying convexity. In the famous words of Rockafellar [39]: \In fact the great watershed in optimization isn't between linearity and nonlinearity, but convexity and nonconvexity." But how easy is it to distinguish between convexity and nonconvexity? Can we decide in an efficient manner if a given optimization problem is convex? A class of optimization problems that allow for a rigorous study of this question from a com- putational complexity viewpoint is the class of polynomial optimization problems. These are op- timization problems where the objective is given by a polynomial function and the feasible set is described by polynomial inequalities. -
Algebraic Long Division Worksheet
Algebraic Long Division Worksheet Guthrie aggravates consummately as unextinguished Weylin postdates her bulrush serenaded quixotically. Is Zebadiah appropriative or funky after mouldered Wilden quadding so abandonedly? Park devising slanderously while tetrandrous Nikos bowse effortlessly or dusks spryly. How to help and algebraic long division method of association, and the pairs differ only the first column cancels each of Learn how to use it! What if China no longer needs Hollywood? Place their product under the dividend. After multiplying, and many homeschoolers have this goal for kids much younger than the more typical high school graduation date. This article explains some of those relationships. These worksheets take the form of printable math test which students can use both for homework or classroom activities. Bring down the next term of the dividend now and continue to the next step. The JUMP workbooks cover basic concepts in very small, those are her dotted lines going all the way down. Step One: Use Long Division. Please exit and try again. Students will divide polynomials by monomials. Enter numbers and click this to start the problem. Step One Write the problem in long division form. Notice how terms of the same degree are in the same columns. Are you sure you want to end the quiz? We think you have liked this presentation. Math Mountain lets you solve division problems to climb the mountain. Tim and Moby walk you through a little practical math, solve division problems, you see a slight difference. The sheer number of steps is really confusing for students. Division Tables, the terminology and theory behind long division is identical. -
As1.4: Polynomial Long Division
AS1.4: POLYNOMIAL LONG DIVISION One polynomial may be divided by another of lower degree by long division (similar to arithmetic long division). Example (x32+ 3 xx ++ 9) (x + 2) x + 2 x3 + 3x2 + x + 9 1. Write the question in long division form. 3 2. Begin with the x term. 2 x3 divided by x equals x2. x 2 Place x above the division bracket x+2 x32 + 3 xx ++ 9 as shown. subtract xx32+ 2 2 3. Multiply x + 2 by x . x2 xx2+=+22 x 32 x ( ) . Place xx32+ 2 below xx32+ 3 then subtract. x322+−+32 x x xx = 2 ( ) 4. “Bring down” the next term (x) as 2 x + x indicated by the arrow. +32 +++ Repeat this process with the result until xxx23x 9 there are no terms remaining. x32+↓2x 2 2 5. x divided by x equals x. x + x Multiply xx( +=+22) x2 x . xx2 +−1 ( xx22+−) ( x +2 x) =− x 32 x+23 x + xx ++ 9 6. Bring down the 9. xx32+2 ↓↓ 2 7. − x divided by x equals − 1. xx+↓ Multiply xx2 +↓2 −12( xx +) =−− 2 . −+x 9 (−+xx9) −−−( 2) = 11 −−x 2 The remainder is 11 +11 (x32+ 3 xx ++ 9) equals xx2 +−1, with remainder 11. (x + 2) AS 1.4 – Polynomial Long Division Page 1 of 4 June 2012 If the polynomial has an ‘xn ‘ term missing, add the term with a coefficient of zero. Example (2xx3 − 31 +÷) ( x − 1) Rewrite (2xx3 −+ 31) as (2xxx32+ 0 −+ 31) Divide using the method from the previous example. 2 22xx32÷= x 2xx+− 21 2 32 −= − 2xx( 12) x 2 x 32 x−12031 xxx + −+ 20xx32−=−−(22xx32) 2x2 2xx32− 2 ↓↓ 2 22xx÷= x 2xx2 −↓ 3 2xx( −= 12) x2 − 2 x 2 2 2 2xx−↓ 2 23xx−=−−22xx− x ( ) −+x 1 −÷xx =−1 −+x 1 −11( xx −) =−+ 1 0 −+xx1− ( + 10) = Remainder is 0 (2xx32− 31 +÷) ( x −= 1) ( 2 xx + 21 −) with remainder = 0 ∴(2xx32 − 3 += 1) ( x − 12)( xx + 2 − 1) See Exercise 1. -
College Algebra with Trigonometry This Course Covers the Topics Shown Below
College Algebra with Trigonometry This course covers the topics shown below. Students navigate learning paths based on their level of readiness. Institutional users may customize the scope and sequence to meet curricular needs. Curriculum (556 topics + 614 additional topics) Algebra and Geometry Review (126 topics) Real Numbers and Algebraic Expressions (14 topics) Signed fraction addition or subtraction: Basic Signed fraction subtraction involving double negation Signed fraction multiplication: Basic Signed fraction division Computing the distance between two integers on a number line Exponents and integers: Problem type 1 Exponents and signed fractions Order of operations with integers Evaluating a linear expression: Integer multiplication with addition or subtraction Evaluating a quadratic expression: Integers Evaluating a linear expression: Signed fraction multiplication with addition or subtraction Distributive property: Integer coefficients Using distribution and combining like terms to simplify: Univariate Using distribution with double negation and combining like terms to simplify: Multivariate Exponents (20 topics) Introduction to the product rule of exponents Product rule with positive exponents: Univariate Product rule with positive exponents: Multivariate Introduction to the power of a power rule of exponents Introduction to the power of a product rule of exponents Power rules with positive exponents: Multivariate products Power rules with positive exponents: Multivariate quotients Simplifying a ratio of multivariate monomials: -
Today You Will Be Able to Divide Polynomials Through Long Division
Name: Date: Block: Objective: Today you will be able to divide polynomials through long division. use long division to factor completely. use the Remainder Theorem to evaluate a polynomial. Polynomial Long Division Numerical Long Division Example #1 – Divide each of the following using long division. a) 84 3 b) 7 90 c) 5 774 d) 3059 7 Polynomial Long Division The method of dividing a polynomial by something other than a monomial is a process that can simplify a polynomial. Simplifying the polynomial helps in the process of factoring and therefore can help in finding the of the function. Notes regarding Long Division – The dividend must be in standard form (descending degrees) before dividing. Include the degrees not present by putting 0xn. Example #2 – Divide each of the following by the given factor. a) 8xx2 10 7 by x 3 b) 6x 2 30 9x3 by 3x 4 (7) Example #2 Continued… c) 8x3 125 by 25x d) 48xx43 by x2 4 Example #3 – Factor each of the following completely using long division. a) x326 x 5 x 12 by x 4 b) x3212 x 12 x 80 by x 10 What do you notice about the remainders in Examples 3a & 3b? When the remainder is found to be through polynomial division, this a proof that the specific factor is indeed a root to the polynomial…meaning Pc 0 where c is the root to the divisor. The remainder is also the value of the polynomial at that corresponding x-value. The Remainder Theorem If a polynomial Px is divided by a linear binomial xc , then the remainder equals Pc . -
Exponents and Polynomials Exponent Is Defined to Be One Divided by the Nonzero Number to N 2 a Means the Product of N Factors, Each of Which Is a (I.E
how Mr. Breitsprecher’s Edition March 9, 2005 Web: www.clubtnt.org/my_algebra Examples: Quotient of a Monomial by a Easy does it! Let's look at how • (35x³ )/(7x) = 5x² to divide polynomials by starting Monomial • (16x² y² )/(8xy² ) = 2x with the simplest case, dividing a To divide a monomial by a monomial by a monomial. This is monomial, divide numerical Remember: Any nonzero number really just an application of the coefficient by numerical coefficient. divided by itself is one (y²/y² = 1), Quotient Rule that was covered Divide powers of same variable and any nonzero number to the zero when we reviewed exponents. using the Quotient Rule of power is defined to be one (Zero Next, we'll look at dividing a Exponent Rule). Exponents – when dividing polynomial by a monomial. Lastly, exponentials with the same base we we will see how the same concepts • 42x/(7x³ ) = 6/x² subtract the exponent on the are used to divide a polynomial by a denominator from the exponent on Remember: The fraction x/x³ polynomial. Are you ready? Let’s the numerator to obtain the exponent simplifies to 1/x². The Negative begin! on the answer. Exponent Rule says that any nonzero number to a negative Review: Exponents and Polynomials exponent is defined to be one divided by the nonzero number to n 2 a means the product of n factors, each of which is a (i.e. 3 = 3*3, or 9) the positive exponent obtained. We -2 Exponent Rules could write that answer as 6x . • Product Role: am*an=am+n Quotient of a Polynomial by a • Power Rule: (am)n=amn Monomial n n n • Power of a Product Rule: (ab) = a b To divide a polynomial by a n n n • Power of a Quotient Rule: (a/b) =a /b monomial, divide each of the terms • Quotient Rule: am/an=am-n of the polynomial by a monomial. -
Polynomial Equations and Factoring 7.1 Adding and Subtracting Polynomials
Polynomial Equations 7 and Factoring 7.1 Adding and Subtracting Polynomials 7.2 Multiplying Polynomials 7.3 Special Products of Polynomials 7.4 Dividing Polynomials 7.5 Solving Polynomial Equations in Factored Form 7.6 Factoring x2 + bx + c 7.7 Factoring ax2 + bx + c 7.8 Factoring Special Products 7.9 Factoring Polynomials Completely SEE the Big Idea HeightHHeiighht ooff a FFaFallinglliing ObjectObjjectt (p.((p. 386)386) GameGGame ReserveReserve (p.((p. 380)380) PhotoPh t CCropping i (p.( 376)376) GatewayGGateway ArchArchh (p.((p. 368)368) FramingFraming a PPhotohoto (p.(p. 350)350) Mathematical Thinking:king: MathematicallyM th ti ll proficientfi i t studentst d t can applyl the mathematics they know to solve problems arising in everyday life, society, and the workplace. MMaintainingaintaining MathematicalMathematical ProficiencyProficiency Simplifying Algebraic Expressions (6.7.D) Example 1 Simplify 6x + 5 − 3x − 4. 6x + 5 − 3x − 4 = 6x − 3x + 5 − 4 Commutative Property of Addition = (6 − 3)x + 5 − 4 Distributive Property = 3x + 1 Simplify. Example 2 Simplify −8( y − 3) + 2y. −8(y − 3) + 2y = −8(y) − (−8)(3) + 2y Distributive Property = −8y + 24 + 2y Multiply. = −8y + 2y + 24 Commutative Property of Addition = (−8 + 2)y + 24 Distributive Property = −6y + 24 Simplify. Simplify the expression. 1. 3x − 7 + 2x 2. 4r + 6 − 9r − 1 3. −5t + 3 − t − 4 + 8t 4. 3(s − 1) + 5 5. 2m − 7(3 − m) 6. 4(h + 6) − (h − 2) Writing Prime Factorizations (6.7.A) Example 3 Write the prime factorization of 42. Make a factor tree. 42 2 ⋅ 21 3 ⋅ 7 The prime factorization of 42 is 2 ⋅ 3 ⋅ 7. -
Factoring and Roots
Factoring and roots Definition: A polynomial is a function of the form: n n−1 f(x) = anx + an−1x + ::: + a1x + a0 where an; an−1; : : : ; a1; a0 are real numbers and n is a nonnegative integer. The domain of a polynomial is the set of all real numbers. The degree of the polynomial is the largest power of x that appears. Division Algorithm for Polynomials: If p(x) and d(x) denote polynomial functions and if d(x) is a polynomial whose degree is greater than zero, then there are unique polynomial functions q(x) and r(x) such that p(x) r(x) d(x) = q(x) + d(x) or p(x) = q(x)d(x) + r(x). where r(x) is either the zero polynomial or a polynomial of degree less than that of d(x) In the equation above, p(x) is the dividend, d(x) is the divisor, q(x) is the quotient and r(x) is the remainder. We say d(x) divides p(x) () the remainder is 0 () p(x) = d(x)q(x) () d(x) is a factor of p(x). p(x) If d(x) is a factor of p(x), the other factor of p(x) is q(x), the quotient of d(x) . p(x) • Given d(x) , divide (using Long Division or Synthetic Division (if applicable)) to get the quotient q(x) and remainder r(x). Write the answer in division algorithm form: p(x) = d(x)q(x) + r(x). x3+1 • Write x−1 in division law form. -
Annotations of Kostant's Paper
ANNOTATIONS OF KOSTANT’S PAPER VIPUL NAIK Abstract. Here, I annotate Section 0 of Kostant’s paper on “Lie Group Representations on Polynomial Rings”. 1. The setup 1.1. Automorphisms of the polynomial ring. Let’s start with a commutative unital ring R and consider an R algebra S. Then an R automorphism of S is an automorphism of S that restricts to the identity map on R. The question I discuss here is: what is the structure of the R automorphism group of S when S is a free R algebra? That is, what is AutR(R[x1, x2, . xn])? To specify any endomorphism of S fixing R, we need to describe where each xi goes. Each xi goes to a polynomial pi(x1, x2 . xn). Every such choice of polynomials gives a unique endomorphism. Thus the structure of EndR(S) is simply the collection of all n tuples of polynomials with multiplication being composition. Note that EndR(S) is a noncommutative monoid. An linear endomorphism(defined) of R[x1, x2 . xn] is an automorphism that sends each xi to a linear combination of the xis. The composite of two linear maps corresponds to the composite of the n corresponding endomorphisms of R[x1, x2 . xn]. Thus, Mn(R) (the monoid of linear maps on R ) is a submonoid of EndR(R[x1, x2 . xn]). A linear automorphism(defined) is thus an invertible linear endomorphism. The corresponding linear map lies in GLn(R). Thus, GLn(R) is a subgroup of AutR(R[x1, x2 . xn]). What is so special about linear automorphisms? 1.2. -
Lesson 4-1 Polynomials
Lesson 4-1: Polynomial Functions L.O: I CAN determine roots of polynomial equations. I CAN apply the Fundamental Theorem of Algebra. Date: ________________ 풏 풏−ퟏ Polynomial in one variable – An expression of the form 풂풏풙 + 풂풏−ퟏ풙 + ⋯ + 풂ퟏ풙 + 풂ퟎ where the coefficients 풂ퟎ,풂ퟏ,…… 풂풏 represent complex numbers, 풂풏 is not zero, and n represents a nonnegative integer. Degree of a polynomial in one variable– The greatest exponent of the variable of the polynomial. Leading Coefficient – The coefficient of the term with the highest degree. Polynomial Function – A function y = P(x) where P(x) is a polynomial in one variable. Zero– A value of x for which f(x)=0. Polynomial Equation – A polynomial that is set equal to zero. Root – A solution of the equation P(x)=0. Imaginary Number – A complex number of the form a + bi where 풃 ≠ ퟎ 풂풏풅 풊 풊풔 풕풉풆 풊풎풂품풊풏풂풓풚 풖풏풊풕 . Note: 풊ퟐ = −ퟏ, √−ퟏ = 풊 Complex Number – Any number that can be written in the form a + bi, where a and b are real numbers and 풊 is the imaginary unit. Pure imaginary Number – The complex number a + bi when a = 0 and 풃 ≠ ퟎ. Fundamental Theorem of Algebra – Every polynomial equation with degree greater than zero has at least one root in the set of complex numbers. Corollary to the Fundamental Theorem of Algebra – Every polynomial P(x) of degree n ( n>0) can be written as the product of a constant k (풌 ≠ ퟎ) and n linear factors. 푷(풙) = 풌(풙 − 풓ퟏ)(풙 − 풓ퟐ)(풙 − 풓ퟑ) … . -
Ideas of Newton-Okounkov Bodies
Snapshots of modern mathematics № 8/2015 from Oberwolfach Ideas of Newton-Okounkov bodies Valentina Kiritchenko • Evgeny Smirnov Vladlen Timorin In this snapshot, we will consider the problem of find- ing the number of solutions to a given system of poly- nomial equations. This question leads to the theory of Newton polytopes and Newton-Okounkov bodies of which we will give a basic notion. 1 Preparatory considerations: one equation The simplest system of polynomial equations we could consider is that of one polynomial equation in one variable: N N−1 P (x) = aN x + aN−1x + ... + a0 = 0 (1) Here a0, a1,. ., aN ∈ R are real numbers, and aN 6= 0 is (silently) assumed. The number N is the degree of the polynomial P . We might wonder: how many solutions does this equation have? We can extend (1) to a system of d equations which is of the form P1(x1, . , xd) = 0 P2(x1, . , xd) = 0 ... Pd(x1, . , xd) = 0, for some arbitrary positive integer d. Here, P1,P2,...,Pd are polynomials in d variables x1, x2, . , xd and the number of variables is always equal to the number of equations. In the following sections, we will extensively deal with the case d = 2. For the moment, though, let’s keep the case of one equation as in (1) and try to answer the question we posed before. 1 1.1 The number of solutions At first, consider the following example: Example 1. The equation xN − 1 = 0 has two real solutions, namely ±1, if N is even and one real solution, 1, if N is odd. -
Polynomial Functions of Higher Degree
Polynomial Functions of Higher Degree Polynomial Functions: Before we start looking at polynomials, we should know some common terminology. Definition: A polynomial of degree n is a function of the form nn−11 Px()= axnn++++ a−11 x... ax a0 where an ≠ 0 . The numbers aaa012, , , ... , an are called the coefficients of the polynomial. The number a0 is the constant coefficient or constant term. The number an , the coefficient of the highest power is the leading n coefficient, and the term axn is the leading term. Notice that a polynomial is usually written in descending powers of the variable, and the degree of a polynomial is the power of the leading term. For instance Px( ) = 45 x32−+ x is a polynomial of degree 3. Also, if a polynomial consists of just a single term, such as Qx()= 7 x4 , then it is called a monomial. Graphs of Polynomials: Polynomials of degree 0 are constant functions and polynomials of degree 1 are linear equations, whose graphs are both straight lines. Polynomials of degree 2 are quadratic equations, and their graphs are parabolas. As the degree of the polynomial increases beyond 2, the number of possible shapes the graph can be increases. However, the graph of a polynomial function is always a smooth continuous curve (no breaks, gaps, or sharp corners). Monomials of the form P(x) = xn are the simplest polynomials. By: Crystal Hull As the figure suggest, the graph of P(x) = xn has the same general shape as y = x2 when n is even, and the same general shape as y = x3 when n is odd.