Fundamental theorem of algebra
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The document discusses several theorems related to polynomials: 1) The Fundamental Theorem of Algebra states that any polynomial of degree greater than 1 with complex coefficients has at least one complex zero. 2) A polynomial of degree n has at most n zeros. 3) A polynomial of degree n greater than 1 with complex coefficients has exactly n complex zeros, if multiplicities are counted.
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Let's think step by step:
1) For x2 + 2x + 5 to be a factor of x4 + px2 + q, when we divide x4 + px2 + q by x2 + 2x + 5, the remainder should be 0.
2) Dividing x4 + px2 + q by x2 + 2x + 5, the quotient will be x2 and the remainder will be p - 5.
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Interpolation techniques - Background and implementation
This document discusses interpolation techniques, specifically Lagrange interpolation. It begins by introducing the problem of interpolation - given values of an unknown function f(x) at discrete points, finding a simple function that approximates f(x).
It then discusses using Taylor series polynomials for interpolation when the function value and its derivatives are known at a point. The error in interpolation approximations is also examined.
The main part discusses Lagrange interpolation - given data points (xi, f(xi)), there exists a unique interpolating polynomial Pn(x) of degree N that passes through all the points. This is proved using the non-zero Vandermonde determinant. Lagrange's interpolating polynomial is then introduced as a solution.
Chapter 3
The document discusses numerical methods for approximating integrals and solving non-linear equations. It introduces the trapezium rule for approximating integrals and provides examples of using the rule. It then discusses iterative methods like the iteration method and Newton-Raphson method for finding approximate roots of non-linear equations, providing examples of applying each method. The objectives are to enable students to use the trapezium rule and understand solving non-linear equations using iterative methods.
Zeros of p(x)
This document discusses various methods for finding the zeros or roots of polynomial functions, including factoring, factor theorem, synthetic division, and using the principle that every polynomial of degree n has n zeros. It provides examples of finding the zeros of polynomials by factorization, using a given zero to find other zeros through synthetic division, and identifying which numbers are zeros of various polynomials. Exercises are included for students to practice finding remaining zeros given one zero and identifying polynomial factors.
Roots equations
S1. Fixed point iteration is a numerical method for solving equations of the form x = g(x) by making an initial guess x0 and repeatedly substituting xn into the right side to obtain xn+1.
S2. The method converges if |g'(α)| < 1, where α is the root and g' is the derivative of g. This ensures the error decreases at each iteration.
S3. Examples show the method can converge rapidly, as in Newton's method, or diverge, depending on the properties of g near the root. Aitken extrapolation can provide a better estimate of the root than the current iterate xn.
Roots equations
S1. Fixed point iteration is a numerical method for solving equations of the form x = g(x) by making an initial guess x0 and repeatedly substituting xn into the right side to obtain xn+1.
S2. The method converges if g(x) is continuous and λ, the maximum absolute value of the derivative of g(x), is less than 1.
S3. Examples show that fixed point iteration can converge slowly if the derivative of g(x) at the root is close to 1, and Aitken's method can be used to accelerate convergence by extrapolating the iterates.
Algebra
The document discusses several topics in algebra including:
1. Indices laws including am x an = am + n, am ÷ an = am - n, and (am)n = amn. Negative and fractional indices are also discussed.
2. Logarithms including the definition that logarithm of 'x' to base 'a' is the power to which 'a' must be raised to give 'x'. Change of base formula is also provided.
3. Series including the definition of finite and infinite series. Notation of sigma notation ∑ is introduced to represent the sum of terms.
2 1 polynomials
The document discusses identifying and evaluating polynomial functions. It defines polynomial functions as functions with whole number exponents and real number coefficients, and no variables in denominators. Polynomials can be written in standard form showing the degree and coefficients. The key points covered are: identifying the degree of a polynomial, finding the zeros of a polynomial function, directly substituting values for x, and using synthetic substitution to evaluate polynomials for any value of x.
polynomial
1) The document discusses polynomials and their properties. It defines what a polynomial is - an expression with terms containing variables and coefficients.
2) It introduces the concept of the degree of a polynomial and defines it as the highest power of the variable terms. Constant polynomials have a degree of zero.
3) The Remainder Theorem is discussed, stating that when dividing a polynomial p(x) by a linear polynomial x-a, the remainder is equal to the value of p(a).
Rational Zeros and Decarte's Rule of Signs
The Rational Zero Theorem provides a method to determine all possible rational zeros of a polynomial function. It states that if p/q is a rational zero, then p is a factor of the constant term and q is a factor of the leading coefficient. Descartes' Rule of Signs can be used to determine the maximum number of positive and negative real zeros by counting the variations in sign of the polynomial function and its substitution of -x. It provides bounds on the number of positive and negative real zeros that are either the number of variations in sign or less by an even integer. The example demonstrates applying these methods to determine all 16 possible rational zeros and the bounds of 0 positive and either 3 or 1 negative real zeros for the given polynomial.
Appendex b
This document provides an overview of solving polynomial equations. It defines polynomials and their key properties like degree, coefficients, and roots. It introduces several theorems for finding roots, including the Remainder Theorem, Factor Theorem, and the idea that a polynomial of degree n has n roots when counting multiplicities. Methods discussed include factoring, long division, and the quadratic formula. The document explains it is not possible to express solutions of polynomials of degree 5 or higher using radicals.
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數學測試
The document discusses mathematical formulas and proofs. It contains:
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2) A claim and proof about the series expansion of (1-x)-3 using binomial theorem.
3) Notations showing equivalence and equality of expressions.
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AJMS_389_22.pdf
This document discusses linear multistep methods for solving initial value differential problems. It makes three key points:
1) Linear multistep methods can be viewed as fixed point iterative methods with a differential operator instead of an integral operator. They converge to a fixed point if consistent and stable.
2) The document proves that any linear multistep method defines a contraction mapping, ensuring convergence to a unique fixed point when the problem is Lipschitz continuous.
3) Examples of explicit and implicit linear multistep methods are given, along with their fixed point iterative formulations. These include Euler's method, trapezoidal method, and Adams-Moulton method.
10.4
Here are the steps to solve each equation by graphing:
1) x2 + 5x + 6 = 0
Write in standard form: x2 + 5x + 6 = 0
Graph the related function y = x2 + 5x + 6. The x-intercepts are the solutions, which are -3 and -2.
2) x2 + 8x + 16 = 0
Write in standard form: x2 + 8x + 16 = 0
Graph the related function y = x2 + 8x + 16. The x-intercepts are the solutions, which are -4 and -4.
3) x2 - 2x + 3 = 0
Write in standard
a) Use Newton’s Polynomials for Evenly Spaced data to derive the O(h.pdf
a) Use Newton’s Polynomials for Evenly Spaced data to derive the O(h4) accurate Second
Centered Difference approximation of the 1st derivative at nx. Start with a polynomial fit to
points at n-2x , n-1x, nx , n+1x and n+2x .
b) Use Newton’s Polynomials for Evenly Spaced data to derive the O(h4) accurate Second
Centered Difference approximation of the 2nd derivative at nx . Remember, to keep the same
O(h4) accuracy, while taking one more derivative than in Part a, we need to add a point to the
polynomial we used in part a.t,s01530456075y,km0356488107120
Solution
An interpolation assignment generally entails a given set of information points: in which the
values yi can,
xi x0 x1 ... xn
f(xi) y0 y1 ... yn
for instance, be the result of a few bodily measurement or they can come from a long
numerical calculation. hence we know the fee of the underlying characteristic f(x) at the set
of points xi, and we want to discover an analytic expression for f .
In interpolation, the assignment is to estimate f(x) for arbitrary x that lies among the smallest
and the most important xi
. If x is out of doors the variety of the xi’s, then the task is called extrapolation,
which is substantially greater unsafe.
with the aid of far the maximum not unusual useful paperwork utilized in interpolation are the
polynomials.
different picks encompass, as an instance, trigonometric functions and spline features
(mentioned
later during this direction).
Examples of different sorts of interpolation responsibilities include:
1. Having the set of n + 1 information factors xi
, yi, we want to understand the fee of y in the
complete c program languageperiod x = [x0, xn]; i.e. we need to find a simple formulation
which reproduces
the given points exactly.
2. If the set of statistics factors contain errors (e.g. if they are measured values), then we
ask for a components that represents the records, and if feasible, filters out the errors.
3. A feature f may be given within the shape of a pc system which is high priced
to assess. In this case, we want to find a characteristic g which offers a very good
approximation of f and is simpler to assess.
2 Polynomial interpolation
2.1 Interpolating polynomial
Given a fixed of n + 1 records points xi
, yi, we need to discover a polynomial curve that passes
via all the factors. as a consequence, we search for a non-stop curve which takes at the values yi
for every of the n+1 wonderful xi’s.
A polynomial p for which p(xi) = yi whilst zero i n is stated to interpolate the given set of
records points. The factors xi are known as nodes.
The trivial case is n = zero. right here a steady function p(x) = y0 solves the hassle.
The only case is n = 1. In this situation, the polynomial p is a directly line described via
p(x) =
xx1
x0 x1
y0 +
xx0
x1 x0
y1
= y0 +
y1 y0
x1 x0
(xx0)
here p is used for linear interpolation.
As we will see, the interpolating polynomial may be written in an expansion of paperwork,
among
these are the Newton shape and the Lag.
Presentation mathmatic 3
The document discusses Chebyshev polynomials. It defines Chebyshev polynomials as orthogonal polynomials related to de Moivre's formula that can be defined recursively. It provides key properties of Chebyshev polynomials including that they are the extremal polynomials for many properties and important in approximation theory. The document also provides formulas for generating Chebyshev polynomials, their orthogonality properties, and their use in representing functions through orthogonal series expansions.
Chapter 11
1) Polynomial coding uses polynomials over Galois fields to encode messages and detect/correct errors in a similar way to binary matrix codes. 2) Encoding a message replaces it with a polynomial modulo an irreducible polynomial. Syndromes reveal the error location. 3) Shift registers provide an efficient way to calculate syndromes by serially shifting the received word and checking for errors.
Algebra Presentation on Topic Modulus Function and Polynomials
This document discusses modulus functions and polynomials. It provides two examples:
1) Solving an absolute value inequality involving a modulus function. The solution is found to be -1 ≤ x ≤ 2.
2) Dividing polynomials. For part a, the remainder and quotient of dividing one polynomial by another is found using long division. For part b, the values of constants a and b are found such that another polynomial divides exactly.
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みなさんこんにちはこれ何文字まで入るの？40文字以下不可とか本当に意味わからないけどこれ限界文字数書いてないからマジでやばい文字数いけるんじゃないの？えこ...
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みなさんこんにちはこれ何文字まで入るの？40文字以下不可とか本当に意味わからないけどこれ限界文字数書いてないからマジでやばい文字数いけるんじゃないの？えこ...
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Fundamental theorem of algebra
- 1. The Fundamental Theorem of Algebra If p(x) is any polynomial of degree n>1 with complex coefficients, then p(x) has at least one complex zero. A related theorem: A polynomial of degree n has at most n zeros. 10 8 6 4 2 -2 -4 -6 -5 5 10 15 10 8 6 4 2 -2 -4 -15 -10 -5 5 10 15 f x( ) = x-2( )2 10 8 6 4 2 -2 -15 -10 -5 5 f x( ) = x-2( )2+3
- 2. Finding Zeros: 14)( 2 +−= xxxp 44)( 2 +−= xxxp a acbb x 2 42 −±− =
- 3. Multiplicity of Zeros: If a polynomial has the same zero more than once, we describe that zero’s multiplicity: Ex: Another theorem: A polynomial of degree n>1 with complex coefficients has exactly n complex zeros, if multiplicities are counted. xxxxp 2510)( 23 ++=
- 4. Complex Zeros xxxxp 54)( 23 ++=
- 8. Factoring Sums and Differences of Powers Recall that can be factored into . What about ? 22 yx − ( )( )yxyx −+ 33 yx −
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