0
votes
1answer
44 views

How do I go about factoring this polynomial?

I am horrid at factoring and I have to find the inflection points of $ f(x)=x^2(x − 3)^3$. So I to find the inflection points I need to set $f'$ equal to $0$ So I have ...
0
votes
0answers
33 views

How can I cleverly use the error term of polynomial interpolation?

Let $f(x):=x^2$. We're interested in the closed form of the error $|I(f)-T_n(f)|$ where ...
1
vote
1answer
35 views

Formula alteration

is there any way to transform the formula$ \frac {1-x}{x-3}$ into something that can be easily sketched, or which will help eliminate $x$ from the denominator?
2
votes
0answers
92 views

Sign of the derivatives of a simple function

Consider the function $f(x)=x^b(1-x)^{1-b}$ defined on $[0,1]$, with $0 < b <1$. How can we prove that the even derivatives $f^{(2k)}$ have a constant sign on $(0,1)$? One can show that this ...
7
votes
1answer
75 views

Find the maximum value of $ \sqrt{x^4-3x^2-6x+13} - \sqrt{x^4-x^2+1} $

If $x\in\mathbb{R}$ find the maximum value of $$ \sqrt{x^4-3x^2-6x+13} - \sqrt{x^4-x^2+1} $$ I tried this: Let $$y= \sqrt{x^4-3x^2-6x+13} - \sqrt{x^4-x^2+1}$$ For maxima ...
1
vote
3answers
41 views

How to establish these two facts about polynomials?

Let $f(x) := \sum_{k=0}^n c_k x^k $ be a polynomial of degree $n\geq 0$ with real coefficeints such that $f(x) = 0$ for $n+1$ distinct real values of $x$. Then how to prove that each $c_k = 0$ and ...
0
votes
2answers
45 views

Prove an inequality (Using Taylor expansion)

Prove: $\frac{x}{1+x} < \ln(1+x) < x$. I thought a good practice would be to prove it using Taylor Expansion. Here's my try: $$\ln(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3}...$$ The n=1 ...
5
votes
1answer
108 views

There cannot exist a rational function $f: \mathbb{R} \to \mathbb{R}$ injective, not surjective

I was looking for a rational function $f: \mathbb{R} \to \mathbb{R}$ that looks like $\arctan$, in that it is injective not surjective well-defined on all $x\in \mathbb{R}$ (no vertical ...
1
vote
1answer
21 views

Cardinality of a set with a recurrence relation.

Let $A = \left\{ f\in \mathbb{N}\rightarrow \mathbb{C} \mid \forall n\in \mathbb{N}. f(n+3) + 3f(n+1) = f(n+2)+f(n) \right\}$ What is $\left|A\right|$? Well, I tried to treat $f$ as a recurrence ...
0
votes
2answers
37 views

Finding a quadratic equation from roots (Vieta's formula)

Given $x_1 = 1-\sqrt 3$ and $x_2=1+\sqrt 3$, What is the quadratic equation? By Vieta's formula: $-\frac{b}{a} = 1-\sqrt 3 + 1+\sqrt 3 = 2$. Hence, $-b = 2a$ $-\frac{c}{a} = (1-\sqrt 3)(1+\sqrt 3) ...
2
votes
2answers
71 views

$f(x) = x^3 - x$ then $f(n)$ is multiple of 3

If $f(x) = x^3 - x$ then $f(n)$ is multiple of 3 for all integer $n$. First i tried $$f(n) = n^3-n=n(n+1)(n-1)\qquad\forall n\ .$$ When $x$ is an integer then at least one factor on the right is ...
6
votes
2answers
186 views

Find the maximum possible value.

For all ordered triples $(p,q,r)$ define the polynomial $$f_{p,q,r}(x)=x^3-px^2+qx-r$$ Let $a_{1},a_{2},a_{3},b_{1},b_{2},b_{3},c_{1},c_{2},c_{3}$ be (not necessarily distinct) positive reals such ...
0
votes
1answer
55 views

Bounds and uniqueness of a transcendental equation

Let $p\in[0,1]$ and $\rho(x): [0,1] \rightarrow [0,\infty)$ such that $$\int_0^1 dx \rho(x) = 1.$$ I'd like to investigate the following transcendental equation: $$\frac{1}{2p} = \int_0^{1} dx ...
3
votes
1answer
80 views

Proving $\cos x < 1 - \frac{x^2}{2} +\frac{x^4}{24}$

I wish to prove the following inequality for $x\ne 0$: $$\cos x < 1 - \frac{x^2}{2} +\frac{x^4}{24}$$ Using the fact that I already prove: $$\cos x > 1 - \frac{x^2}{2}$$ My try: $\cos x = 1 - ...
0
votes
0answers
48 views

Indefinite integration of general polynomial. Is this correct?

I was reading some notes of a guy I was tutoring the other day on basic calculus. He noted that if $$\int{x^n dx}=\frac{x^{n+1}}{n+1}+c,$$ then that can be extrapolated to all polynomials. He wrote, ...
0
votes
0answers
10 views

Learning a multivariate polynomial with dependent coefficients

I have a polynomial of the form of $ K^2((a-i)^2 + (b-j)^2 + c^2) = (ct)^2$ where $a,b,c,t$ are unknowns. I have multiple observation points for the values of $i,j,K$. Can I use some technique to ...
1
vote
1answer
20 views

relation between the number of real roots of the derivative and the original polynomial

If the derivative of polynomial has n real roots then can we conclude that the original polynomial has to have n+1 real roots?
0
votes
1answer
36 views

conclusion about roots for positive derivative of a polynomial

If the derivative of a polynomial is always positive then what can we conclude about the number of real roots the original polynomial?
1
vote
3answers
70 views

Relation between the roots of $x^2+x+1$ and its derivative

If $f(x)$ is a polynomial in n degree and has $n$ real roots then is it necessary that $f'(x)$ has to have $n-1$ real roots? If this is so then $x^2+x+1$ has no real roots but the derivative of the ...
1
vote
2answers
78 views

Using telescoping property to prove difference of powers

Ok so I have started working through Apostol calculus and as you can see I am stuck. The problem is that I can not see the telescoping pattern anywhere for following problem. Prove that $$a^n - b^n ...
1
vote
3answers
40 views

Inequality for quartic polynomial depending on a parameter

Let $f(x) = \frac 14 x^4 - \frac \alpha2 x^2 - (\alpha-1) x - \frac \alpha 2 + \frac 3 4 $. I want to show that there exists an $\alpha>1$ such that $f(x)\geq 0 $ for $x\leq 0$. Even more, it ...
1
vote
4answers
68 views

Constructing a polynomial with certain zeroes.

I want to construct a polynomial $f(x)$ that has zeroes at $-9,\,-5,\,0,\, 5,\, 9$. Can somebody provide a method (or perhaps some hints) for solving this?
2
votes
1answer
260 views

How to make this polynomial the zero polynomial?(recursively)?

Given a fixed $\beta \in \mathbb{R}$, I want to find the $c_0,...,c_n$ for arbitrary $n \in \mathbb{N}$ such that the polynomial \begin{align}P_n(z):=z(1-z) ...
1
vote
1answer
42 views

Find the Taylor polynomial of degree 4 for cos(x), for x near 0

I am self studying calculus and I need help solving a Taylor Series problem. 1a) Find the Taylor polynomial of degree 4 for cos(x), for x near 0: I think the answer would be: ...
1
vote
1answer
42 views

Question on the norms

I got stuck with the following (simple) question since the result I got seems to be counterintuitive: I have a function defined in terms of its Chebyshev expansion, i.e. ...
0
votes
0answers
32 views

Finding roots of a fractional exponential equation.

If we consider a polynomial equation its easy to find the number of roots associated with the expression by applying Descartes Rule. This method, however, doesn't work with non integer exponents. ...
2
votes
4answers
73 views

Limit of a rational function to the power of x

Ok so I have been trying for days already to find a solution to this all around the web and in math books but to no success. The problem is to evaluate a limit of a function composed by polynomial ...
4
votes
2answers
74 views

Is limits $\lim_{x\to\infty}{(x-2)^2\over2x+1}=\dfrac{1}{2}$?

$$\lim_{x\to\infty}{(x-2)^2\over2x+1}=\dfrac{1}{2}$$ I used an online calculator and it said it was actually $=\infty$ Here's how I calculate it: ...
0
votes
1answer
25 views

Use economisation to find linear approximation to x^2-x-1?

I've been given the solution to this question... It uses chebychev, and you get: $1/2(2x^2-1)-2x-1/2$ So the Chebyshev economisation polynomial is $-2T1 -1/2 T0$ I can see the logic in how this ...
4
votes
2answers
142 views

Proof polynomial has only one real root.

I need to prove that this polynomial equation: $$x^5-(3-a)x^4+(3-2a)x^3-ax^2+2ax-a=0\quad\text{ for }\quad a\in(0,\frac{1}{2}).$$ has only one root. That it has one real root is obvious because it is ...
23
votes
1answer
524 views

An awful identity

We take place on $\mathbb C(x_1,...,x_r,x'_1,...,x'_p,u_0,...,u_r,u'_0,...,u'_p)$ with $r,p\in \mathbb N$ Show that : $$\displaystyle{\sum_{i=1}^r \left( \frac{\prod_{j=0}^r (u_j-x_i) ...
1
vote
2answers
52 views

Describing asymptotic behaviour of a function

For question B! x^2+x+1/x^2 = 1+ [x+1/x^2] shouldnt the answer be asymptote at x=0 and y=1 ?? i dont understand the textbook solution
5
votes
1answer
94 views

Analyzing a fourth degree polynomial

Let $a,b$ and $c$ be real numbers. Then prove that the fourth degree polynomial in $x$ $acx^4+b(a+c)x^3+(a^2+b^2+c^2)x^2+b(a+c)x+ac$ has either 4 real roots or 4 complex roots. I have never solved a ...
8
votes
2answers
301 views

Real roots of a polynomial

Let $p$ be an even degree polynomial with real coefficients such that the product of the constant term and the leading coefficient is negative. Show that $p$ has at least two real roots. Thanks!
1
vote
1answer
67 views

A Cubic Equation

$2x^3+ax^2+bx+4=0$, $(a,b \in R^+)$ has three real roots. Then : A. $a\geqslant 4.2^{\frac 1 3}$ B. $a\geqslant 1.2^{\frac 1 3}$ C. $a\geqslant 6.2^{\frac 1 3}$ D. $a\geqslant 2.2^{\frac 1 3}$ ...
0
votes
1answer
45 views

Bounds on coefficients of close polynomials

I've got two polynomials $p, \hat{p}:\mathbb{R}^2\rightarrow \mathbb{R}$ of degree $2\times2\ $ which are close together around $0$: $$|p(\mathbf{x})-\hat{p}(\mathbf{x})|<\varepsilon \quad \forall ...
1
vote
0answers
131 views

Solving an 8th degree polynomial

I know that through the Abel Ruffini Theorem the general solution to a polynomial of degree five or more cannot be found explicitly. But are there are any other ways to find the roots of such a ...
6
votes
1answer
82 views

Show that $P$ is divided to simple roots knowing that $a_{k}^2-4a_{k-1}a_{k+1}>0$

Let $P(X)=a_0+a_1X+..+a_nX^n\in R[X]$ Assume that $\forall k, a_k>0$ and $a_{k}^2-4a_{k-1}a_{k+1}>0$ Show that $P$ is divided to simple roots in $R[X]$. i.e. ...
0
votes
1answer
40 views

Inferring a characteristic of a ratio of functions from the ratio of their derivatives

This is a strange one, but I need help trying to understand whether there is any logic behind this or not. Given $\frac {f(\sqrt{2})}{g(\sqrt{2})}=2$, and $\frac {f'(x)}{g'(x)}>2$ for all ...
14
votes
6answers
457 views

Proving $x^{4}+x^{3}+x^{2}+x+1$ is always positive for real $x$

So I was bored in class and decided to graph polynomials in geogebra, I noticed that $x^{4}+x^{3}+x^{2}+x+1$ and $x^{6}+x^{5}+x^{4}+x^{3}+x^{2}+x+1$, are all above the x-axis. Now I am wondering if ...
0
votes
1answer
38 views

Leading coefficient of a polynomial quotient is equal

Isn't there something special that you can infer when a polynomial's leading coefficients are equal on the numerator and denominator, like this: $\frac{x^4}{x^4+3x^4}$ I'm trying to find the limit ...
0
votes
1answer
40 views

How to simplify floor polynomial given lower bound on x?

$$ \left\lfloor\frac{8x^2 + 5x -4}{3x^2 + x}\right\rfloor $$ where $x$ > $\sqrt{8}$ How would you simplify this type of expression? *Please note the floor operation surrounding the expression ...
0
votes
3answers
58 views

How to generically solve polynomial expressions given a minimum value of x?

Given something like $\dfrac{8x^2 + 2x + 7}{3x^2 + 2x}$, and $x > \sqrt8$, what strategy would you employ to simplify this expression?
1
vote
1answer
84 views

Why can't a non-zero polynomial satisfy some equations?

I'm having a hard time visually picturing/understanding how to explain why a non-zero polynomial function cannot satisfy the equation: $f''(x)$ = $-f(x)$ So is it basically asking to explain why a ...
3
votes
2answers
1k views

Proof that every polynomial of odd degree has one real root

I want to prove that every real polynomial of odd degree has at least one real root, using the intermediate value theorem. Let $P(x) = x^{2n+1} + a_n x^{2n} + . . . + a_0$ for each $a_i \in ...
0
votes
2answers
61 views

Spivak Calculus problem with inductive proof of polynomial property

Ok so I have been trying to do this proof of what seems to be a version of Factor theorem(I am now well versed in math so please forgive me if I am wrong). Problem goes like this: Prove that ...
6
votes
3answers
80 views

$p(x)\geq 0 \forall x\Rightarrow p(x)+p'(x)+p''(x)+…+p^{(n)}(x)\geq 0$ [duplicate]

$p(x)\geq 0 \forall x \in \mathbb{R} \Rightarrow p(x)+p'(x)+p''(x)+...+p^{(n)}(x)\geq 0$, where p(x) is a polynomial of degree n. I showed: $a_{n}+...+a_{0}\geq 0$, ...
1
vote
4answers
96 views

If something holds for n+1 distinct values then it holds for all values.Proving a property of polynomial

Ok I am stuck on proving a property of polynomial.It basically goes like this. If $$f(x) = \sum_{k=0}^{n} c_kx^k $$ is equal to zero for n+1 distinct real values x,then f(x) is equal to 0 for all ...
3
votes
3answers
131 views

Solve $t^4+4 t^3+6 t^2+4 t-32 t^{1/4}+1 = -16 $

I'm trying to solve the following equation: $$(t+1)^4 - 32 t^{\frac{1}{4}}=-16 $$ where t $\geq 0$, which is equivalent to $$t^4+4 t^3+6 t^2+4 t-32 t^{\frac{1}{4}}+1 = -16 $$ Wolfram Alpha tells that ...
1
vote
0answers
91 views

Is this a correct way to prove what the derivative of a polynomial function is?

After trying a polynomial long division problem with a lot of wondering how to go about answering it I proceeded by most likely overcomplicating things but the equation derived seems to work at ...