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- '''THEOREM.''' A linear polynomial ...h $f(x+T)$ and work our way to $f(x)$, identically. For example, no linear polynomial $f(x)=mx+b,\ m\ne 0$, is periodic:143 KB (24,052 words) - 13:11, 23 February 2019
- '''Definition.''' The ''characteristic polynomial'' of an $2\times 2$ matrix $A$ is defined to be: *the eigenvalues are the real roots of the (quadratic) characteristic polynomial $\chi_A$, and, therefore,113 KB (18,750 words) - 02:33, 10 December 2018
- ...lar value of $x$, we can find a special analog of the standard form of the polynomial for each $x=a$. The polynomials are still sums of powers just not of $x$ bu '''Proposition.''' For each real number $a$, every degree $n$ polynomial $P$ can be represented in the form ''centered at $x=a$'', i.e.,113 KB (19,100 words) - 23:07, 3 January 2019
- '''Example.''' To find the inverse of a linear polynomial A linear polynomial,142 KB (23,566 words) - 02:01, 23 February 2019
- Any polynomial can be built from $x$ and constants by multiplication and addition. Therefo [[image:cubic polynomial.png| center]]107 KB (18,743 words) - 17:00, 10 February 2019
- *Suppose $f$ is a polynomial of degree $55$ and its leading term is $-1$. Describe the long term behavio *For the polynomial $f(x)=-2x^2(x+2)^2(x^2+1)$, find its $x$-intercepts.17 KB (2,933 words) - 19:37, 30 July 2018
- '''MTH 130 College Algebra.''' 3 hrs. Polynomial, rational, exponential, and logarithmic functions. Graphs, systems of equat * 2.4 Higher Degree Polynomial Equations10 KB (1,078 words) - 19:07, 16 December 2016
- Any polynomial can be built from $x$ and constants by multiplication and addition. Therefo [[image:cubic polynomial.png| center]]51 KB (9,271 words) - 20:02, 8 September 2016
- In other words, our “function of functions” has the same property as a linear polynomial: *The derivative of a constant polynomial is zero:82 KB (14,116 words) - 19:50, 6 December 2018
- The analysis starts with the characteristic polynomial: Recall that the characteristic polynomial of matrix $F$ is63 KB (10,958 words) - 14:27, 24 November 2018
- We know from Chapter 3 that under a linear polynomial $f(x)=mx+b$ with $m>0$, the distance is increased by a factor of $m$ or dec *the derivative of a quadratic polynomial is linear, and75 KB (13,000 words) - 15:12, 7 December 2018
- 3. Polynomial and Rational Functions 3.2 [[Polynomial]] Functions2 KB (269 words) - 18:53, 16 November 2011
- *Estimate the coefficients of the Taylor polynomial $T_1$ of order $1$ centered at $a=1$ of the function $f$ shown above. Provi *What degree Taylor polynomial one would need to approximate $e^{.01}$ within $.001$? (Answers may vary an15 KB (2,591 words) - 17:15, 8 March 2018
- The ''characteristic polynomial'' of matrix $A$ is This is a polynomial of $\lambda$!12 KB (1,971 words) - 01:09, 12 October 2011
- '''MTH 130 College Algebra.''' 3 hrs. Polynomial, rational, exponential, and logarithmic functions. Graphs, systems of equat Chapter 5. Polynomial and Rational Functions6 KB (752 words) - 04:19, 13 December 2013
- ''MTH 130 College Algebra.'' 3 hrs. Polynomial, rational, exponential, and logarithmic functions. Graphs, systems of equat Chapter 5. Polynomial and Rational Functions6 KB (850 words) - 16:52, 29 November 2014
- 5. Polynomial and Rational Functions 5.1 Polynomial Functions and Models3 KB (349 words) - 16:29, 8 August 2013
- ...lus with Scientific Applications.''' Functions used in calculus including polynomial, rational, exponential, logarithmic, and trigonometric. Systems of equation * 4. POLYNOMIAL AND RATIONAL FUNCTIONS.7 KB (890 words) - 16:32, 20 April 2016
- Its [[Taylor polynomial]] of degree $3$ is Consider the quadratic polynomial of $t$:14 KB (2,404 words) - 15:04, 13 October 2011
- #$r$ is a simple [[root]] of the [[characteristic polynomial]] of A and its [[eigenspace]] is 1-dimensional (the geometric multiplicity #r is a simple root of the characteristic polynomial of A and its [[eigenspace]] is 1-dimensional (the geometric multiplicity of2 KB (239 words) - 15:08, 25 August 2011
- 16 polynomial rings Zeros of an irreducible polynomial 3625 KB (568 words) - 15:23, 16 November 2011
- *$y = mx + b$, the best affine approximation, linear polynomial; *$ax^2 + bx + c$, quadratic polynomial.3 KB (466 words) - 15:43, 31 March 2013
- *$y = mx + b$ the best affine approximation, linear polynomial; *$ax^2 + bx + c$, quadratic polynomial.34 KB (5,665 words) - 15:12, 13 November 2012
- *the derivative of a linear polynomial is constant, but are the linear polynomials the only functions with this pr *the derivative of a quadratic polynomial is linear, but are the quadratic polynomials the only functions with this p84 KB (14,321 words) - 00:49, 7 December 2018
- $\bullet$ '''7.''' For the polynomial $f(x)=-2x^2(x+2)^2(x^2+1)$, find its $x$-intercepts. $\bullet$ '''10.''' Find a formula for a polynomial with these roots: $1$, $2$, and $3$.2 KB (308 words) - 17:21, 2 March 2016
- ...here may still be exceptions that will call for using the ''cubic'' Taylor polynomial $T_3$ of $f$. And so on. We will need all the Taylor polynomials, i.e., the64 KB (11,426 words) - 14:21, 24 November 2018
- '''Theorem (Limits of Polynomials).''' Suppose we have a polynomial of degree $p$ with the leading coefficient $a_p\ne 0$. Then the limit of th So, as far as its behavior at $\infty$, for a polynomial,64 KB (10,809 words) - 02:11, 23 February 2019
- #Find the Taylor polynomial of order 2 centered at a=1 of the function $f(x)=e^{x²}$. #Find the Taylor polynomial $T_{2}(x)$ of order $2$ centered at $a=\pi$ of the function $f(x)=\sin^{2}x4 KB (567 words) - 20:23, 13 June 2011
- Since this holds for all x, this [[polynomial]] is the zero polynomial:27 KB (4,667 words) - 01:07, 19 February 2011
- $\bullet$ '''4.''' Estimate the coefficients of the Taylor polynomial $T_1$ of order $1$ centered at $a=1$ of the function $f$ shown above. Provi $\bullet$ '''7.''' What degree Taylor polynomial one would need to approximate $e^{.01}$ within $.001$? (Answers may vary an1 KB (226 words) - 19:13, 13 October 2014
- Since this holds for all $x$, this [[polynomial]] is the zero polynomial:26 KB (3,993 words) - 19:48, 26 August 2011
- 10 The graph of a quadratic polynomial is a parabola 5 Polynomial functions16 KB (1,933 words) - 19:50, 28 June 2021
- $\bullet$ '''7.''' For the polynomial $f(x)=-2x^2(x+2)^2(x^2+1)$, find its $x$-intercepts. $\bullet$ '''10.''' Find a formula for a polynomial with these roots: $1$, $2$, and $3$.2 KB (308 words) - 15:23, 2 March 2016
- [[image:graph of a polynomial.png| center]] ...d numbers $m$ and $b$. When $m\ne 0$, such a function is called a ''linear polynomial''.151 KB (25,679 words) - 17:09, 20 February 2019
- '''Theorem (Fundamental Theorem of Algebra).''' Every non-constant (complex) polynomial has a root. '''Proof.''' Choose the polynomial $p$ of degree $n$ to have the leading coefficient $1$. Suppose $p(z)\ne 0$46 KB (7,846 words) - 02:47, 30 November 2015
- '''Theorem (Fundamental Theorem of Algebra).''' Every non-constant (complex) polynomial has a root. '''Proof.''' Choose the polynomial $p$ of degree $n$ to have the leading coefficient $1$. Suppose $p(z)\ne 0$45 KB (7,738 words) - 15:18, 24 October 2015
- $\bullet$ '''10.''' Suppose $f$ is a polynomial of degree $55$ and its leading term is $-1$. Describe the long term behavio2 KB (389 words) - 19:13, 6 May 2016
- *The formula for the Taylor polynomial of $n$th degree of function $f$ at $a\in {\bf R}^N$ is: $$T_n(x)=\sum_{|\al14 KB (2,538 words) - 18:35, 14 October 2017
- ...simpler functions. It's similar to the [[Taylor series]] but instead of [[polynomial]]s it uses [[trigonometric functions]].2 KB (405 words) - 15:03, 17 February 2011
- #Find the Taylor polynomial of order 2 centered at $c=π/2$ of the function $f(x)=\sin ²x$. How accura6 KB (823 words) - 20:23, 13 June 2011
- where \( P \) and \( Q \) are [[polynomial]]s.19 KB (2,850 words) - 15:04, 19 March 2011
- &= e^{-x^{2}} \underbrace{( -2x + 4x^{3} - 4x)}_{\text{cubic polynomial}} \\49 KB (8,436 words) - 17:14, 8 March 2018
- '''MTH 130 College Algebra.''' 3 hrs. Polynomial, rational, exponential, and logarithmic functions. Graphs, systems of equat3 KB (355 words) - 22:00, 12 December 2011
- #Find the Taylor polynomial of order $2$ centered at $c=\pi/2$ of the function $$f(x)=\sin^{2}x.$$ How2 KB (318 words) - 20:24, 13 June 2011
- ...Find the [[trace]] ${\rm Tr \hspace{3pt}}A$ from from the [[characteristic polynomial]] $\chi_A$.4 KB (677 words) - 17:31, 13 October 2011
- #For the polynomial $f(x)=2(x-2)^2(x+3/7)^3$, find its $x$-intercepts and its large scale behav2 KB (321 words) - 03:09, 6 November 2018
- 8. Find the Taylor polynomial of degree $4$ that would help to approximate $e^{1.01}$.990 bytes (151 words) - 17:15, 12 October 2012
- ...roblems are hard, especially when the graph is large. Most of them are non-polynomial. Therefore, reducing the size of a graph means that we can save much time t11 KB (1,674 words) - 23:20, 25 October 2011
- $\bullet$ '''4.''' For the polynomial $f(x)=-2x(x-2)^2(x+1)^3$, find its $x$-intercepts and its large scale behav2 KB (410 words) - 03:09, 6 November 2018
- $\bullet$ '''7.''' What degree Taylor polynomial one would need to approximate $\sin (-.01)$ within $.001$? Explain the form2 KB (312 words) - 16:00, 14 December 2014
