Polynomials are expressions like $15x^3 - 14x^2 + 8$. Questions tagged with this concern common operations on polynomials, like adding, multiplying, polynomial long division, factoring and solving for roots.

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344 views

Prove that ideal generated by… Is a monomial ideal

Similar questions have come up on the last few past exam papers and I don't know how to solve it. Any help would be greatly appreciated.. Prove that the ideal of $\mathbb{Q}[X,Y]$ generated by ...
3
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3answers
88 views

Find $p$ if $(x + 3)$ is a factor of $x^3 - x^2 + px + 15$.

I'm just making sure I answered this correctly. If $(x+3)$ is a factor, then $P(-3)$ would equal $0$, correct?
0
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4answers
272 views

How to graph the equation: $y=\frac {x-2}{x+1}$?

the title says it all. I'm pretty sure this is a hyperbola, but is there an alternative way of doing this besides a table of values? "Graph the equation $y=\frac {x-2}{x+1}$" I know that $x$ cannot ...
2
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2answers
108 views

Help to understand the ring of polynomials terminology in $n$ indeterminates

In the Hungerford's book, page 150, the author defines a ring of polynomials in "n" indeterminates in the following manner: After the author defines the operations in this ring with a theorem: ...
2
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1answer
56 views

For the following monic polynomial,$f$ of even degree how to prove that that $lim_{|x|\to\infty }(\sqrt {f(x)}-g(x))=0$

For any monic polynomial $f \in \mathbb {Q[x]}$ of even degree,how to prove, there exists polynomial $g \in \mathbb {Q[x]}$ such that $lim_{|x|\to\infty }(\sqrt {f(x)}-g(x))=0$
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2answers
176 views

What am I doing wrong in these quartic formula calculations?

I was a bit surprised that there is a general formula for the roots of a quartic equation, so I decided to test Wikipedia's version of it myself. To my surprise, I have arrived at a correct answer ...
1
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1answer
96 views

Finding the value of a coefficient given the roots are equal?

I've had some trouble with this question: "$P(x)$ denotes the quadratic polynomial $kx^2+(k-1)x-(2k-1)$, where $k$ is a rational, real number. Find the value of $k$ for which the roots of $P(x)=0$ ...
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3answers
100 views

find out the value of $\dfrac {x^2}{9}+\dfrac {y^2}{25}+\dfrac {z^2}{16}$

If $(x-3)^2+(y-5)^2+(z-4)^2=0$,then find out the value of $$\dfrac {x^2}{9}+\dfrac {y^2}{25}+\dfrac {z^2}{16}$$ just give hint to start solution.
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2answers
457 views

Deducing a coefficient from a cubic polynomial given a divisor and remainder?

I got this question which I don't understand: "Suppose $x^3 - 2x^2 + a = (x + 2) Q(x) + 3$ where $Q(x)$ is a polynomial. Find the value of a." I know the identity: $P(x)=A(x)Q(x)+R(x)$, but I'm not ...
2
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0answers
102 views

A question about cubic equation.

I'd like to share my doubt on cubic equation. Step 1: $ax^3+bx^2+cx+d=0$, Step 2: We can substitute $x=y-\frac b {3a}$ to get $y^3+py+q=0$ where $p,q$ are something. Step 3: By Vieta's ...
4
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2answers
92 views

Simplifying this expression $(e^u-1)(e^u-e^l)$

Is it possible to write the following $$(e^u-1)(e^u-e^l)$$ as $$e^{f(u,l)}-1?$$
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votes
3answers
68 views

Factorize in R[x]

I have the polynomial $x^8+1$, I know that there's no root for solve this in $\Bbb R[x]$ but i want to factorize this to the minimal expression. This is possible or this is irreducible?
4
votes
5answers
630 views

Polynomials - The sum of two roots

If the sum of two roots of $$x^4 + 2x^3 - 8x^2 - 18x - 9 = 0$$ is $0$, find the roots of the equation
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4answers
135 views

Irreducible Polynomial in $\mathbb F_{256}$.

Let $\mathbb F_{256}$ be the finite field with $2^8 = 256$ elements. Consider the polynomial over this field $$ x^2 + x + 1. $$ I wanted to know if it is irreducible, so I calculated it for all ...
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1answer
28 views

(Follow Up) Checking the solutions of a quadratic polynomial

I'm following up from this question: Solve a polynomial involving geometric progression? I have had trouble with this question: "Solve the equation $8x^3−38x^2+57x−27=0$" if the roots are in ...
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2answers
868 views

Solve a polynomial involving geometric progression?

I have had trouble with this question: "Solve the equation $8x^3 - 38x^2 + 57x -27 = 0$" if the roots are in geometric progression. Any help would be appreciated.
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2answers
181 views

How to prove two polynomials have no zeroes in common?

The question asked: Divide the polynomial $P(x) = x^3 + 5x^2 - 22x - 6$ by $G(x) = x^2 - 3x + 2$. I did, and got the answer: $(x+8)(x^2-3x+2)-22$. However, it now asks to: "Show that $P(x)$ and ...
2
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3answers
919 views

How to form a cubic equation with the substitution method?

I had this question: "Find the cubic equation whose roots are twice the roots of the equation $3x^3 - 2x^2 + 1 = 0$" In my first attempt, I solved it through the use of simultaneous equations, where ...
1
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1answer
24 views

solving for one variable in terms of others

A question from Steward's Precalculus textbook 5th, Pg 55, the original formula is $$h=\frac{1}{2}gt^2+V_0t$$ the question asks to write the formula in terms of $t$, the answer is ...
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0answers
54 views

Find the factorization of the polynomial as a product of irreducible [duplicate]

Find the factorization of the polynomial $x^5-x^4+8x^3-8x^2+16x-16$ as a product of irreducible on rings $R[x]$ and $C[x]$ Testing with the simplest possible root in this case, $P(1)=0$ Applying the ...
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0answers
79 views

How to use two number to form a Jones polynomial

According to the Wikipedia article on Knots, The number of crossing (rule $1$) and a line crossing the triangle (rule $2$) form a number such as $3,1$. With these two numbers, how do you form a ...
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6answers
333 views

Cubing a simple thing

I am trying to expand $\quad (x + 2)^3 $ I am actually not to sure what to do from here, the rules are confusing. To square something is simple, you just foil it. It is easy to memorize and execute. ...
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1answer
118 views

Find the factorization of the polynomial as a product of irreducible on rings R[x] and C[x]

Find the factorization of the polynomial $x^5-x^4+8x^3-8x^2+16x-16$ as a product of irreducible on rings $\Bbb R[x]$ and $\Bbb C[x]$ Testing with the simplest possible root in this case, $P(1) = 0$ ...
3
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2answers
54 views

Deducing a coefficient from a cubic polynomial?

I fully answered the question, and got that $k=-3$, but the answer says it's positive. Can anyone show me my mistake? "Given that $x-2$ is a factor of the polynomial $x^3 - kx^2 - 24x + 28$, find $k$ ...
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1answer
244 views

Calculating in quotient ring of $\mathbb{R}[X]$

Part of an old Oxford exam (1992 A1) We want to find which elements of the quotient ring $\mathbb{R}[X]/(x^3-x^2+x-1)$ are equal to their own square. Now, we note first that ...
4
votes
2answers
112 views

How to show that there does not exist any integer $b$ with $f(b)=14.$

Let $f(x)$ be a polynomial with integer coefficients. Suppose that there exist distinct integers $a_1,a_2,a_3,a_4,$ such that $f(a_1)=f(a_2)=f(a_3)=f(a_4)=3.$ Then show that there does not exist ...
2
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1answer
57 views

Minimal Polynomials Annihilating an Abelian Torsion-Free Group

Let $A$ be an abelian torsion-free group. Let $\theta \in\operatorname{Aut}A$. Assume that $\theta$ has a finite period in $\operatorname{Aut} A$, say $n$. Obviously $\theta^n-1$ annihilates $A$ (i.e. ...
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1answer
56 views

Polynomials -finding sum of symmetric function of cubic polynomial

Problem : If $\alpha , \beta,\gamma$ are the roots of $x^3+bx+c=0$ then $\alpha^2\beta +\alpha \beta^2+\beta^2\gamma +\beta \gamma^2+\gamma^2 \alpha+\gamma \alpha^2$ is equal to Options are : a) ...
0
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1answer
457 views

Coefficients of Newton interpolation polynomial

Given distinct $y_0,...,y_m$ in $\mathbb R$, let $N_m(x)$ be the Newton interpolation polynomial of degree $m$. That is, $N_m(x) = \sum_{n=0}^{m}a_nw_n(x)$ where $w_0 = 1$, $w_n(x) = ...
12
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3answers
720 views

Irreducibility of $x^n-x-1$ over $\mathbb Q$

I want to prove that $p(x):=x^n-x-1 \in \mathbb Q[x]$ for $n\ge 2$ is irreducible. My attempt. GCD of coefficients is $1$, $\mathbb Q$ is the field of fractions of $\mathbb Z$, and $\mathbb Z$ ...
0
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0answers
28 views

Number of roots to system of Polynomials

If you have a system of k polynomials of dimension k and degree r is the number of solutions equal to: k^s? This appears to be the pattern and intuitively one could argue that each of the s systems ...
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2answers
608 views

Can you help me reverse the Minimum Curvature Method?

The minimum curvature method is used in oil drilling to calculate positional data from directional data. A survey is a reading at a certain depth down the borehole that contains measured depth, ...
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1answer
186 views

Bounding the roots of the sum of two monic polynomials with real coefficients.

Let $P_1(z)$ and $P_2(z)$ be monic polynomials with real coefficients and roots $\{z_1^{(1)},z_1^{(2)},...\}$ and $\{z_2^{(1)},z_2^{(2)},...\}$, respectively. Are there any results relating the ...
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2answers
360 views

Zeros of a cubic polynomial with rational coefficients

While discussing a related problem, one of my friends came out with a question as follows: Is it possible that a cubic polynomial $p(x) \in \Bbb{Q}[x]$ has all of its zeros to be both real and ...
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2answers
194 views

What is the general equation of a cubic polynomial?

I had this question: "Find the cubic equation whose roots are the the squares of that of $x^3 + 2x + 1 = 0$" and I kind of solved it. In that my answer was $x^3 - 4x^2 + 4x + 1$, but it was actually ...
2
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1answer
93 views

Is there a geometric relationship between plane geometry and polynomials?

It is well known that the complex plane is algebraically closed: Every polynomial has a zero. The relationship seems, to me, to run deeper: For every complex-differentiable function, there exists a ...
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2answers
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A simple proof about $e^x$?

Do you guys think this is correct? I am trying to prove that there is no single-term polynomial function (oxymoron, I know) $f(x)$ which is always (or at least as x approaches infinity) greater than ...
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1answer
62 views

Prove (without quoting any theorems) that polynomials on [0,1] are continous

I'm confused as to go about this problem. I feel as if we have to show that $P [0,1] \in C^{0}[0,1]$ by letting $f = a_{n}x^{n} + a_{n-1}x^{n-1} + .... + a_{1}x^{1} + a_{0}$ We must show that ...
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0answers
58 views

A Nonzero Alternating Bilinear Form on the Space $P_1(F)$ Over $F$

Can anybody think of an example of a nonzero alternating bilinear form on the space $P_1(F)$ over $F$. $F$ is a general field like $\mathbb{R}$ or $\mathbb{C}$. $P_1(F)$ is the set of all ...
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3answers
215 views

Solve $(x^2 + 5)^2 - 15(x^2 + 5) + 54 = 0$

I got the square root of 14 and 11 but the answer book states that these answers are wrong. Can someone help me? I used this formula to find the individual roots $x = -\frac{p}{2} \pm ...
2
votes
2answers
234 views

Solve the roots of a cubic polynomial?

I have had trouble with this question - mainly due to the fact that I do not fully understand what a 'geometric progression' is: "Solve the equation $x^3 - 14x^2 + 56x - 64 = 0$" if the roots are in ...
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3answers
113 views

Solve a cubic equation? [duplicate]

Need help with solving an equation: Solve the equation $5x^3 - 24x^2 + 9x + 54 = 0$ given that two of it's roots are equal. Any help would be greatly appreciated. Thanks!
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1answer
29 views

Relationship between 2 Dimensional Quadratic systems and roots

Given four points $(x_1, y_1) (x_2, y_2) (x_3, y_3) (x_4, y_4)$ How does one construct a system of two equations: $a_1x + a_2x^2 + a_3y + a_4y^2 + a_5xy = c_1$ $b_1x + b_2x^2 + b_3y + b_4y^2 + ...
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1answer
58 views

Irreduciblity of polynomial in $\Bbb Q[x]$

I am trying to prove that $x^5 + 16$ is irreducible in $\Bbb Q[x]$. (Hint: Consider $(x-1)^5 + 16$) Not sure how to use the hint, thanks for any help in advance.
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1answer
394 views

Test for polynomial reducibility with binary coefficients

I'm learning about Galois Fields, in particular $GF(2^8)$, as they are applied to things like the AES algorithm and Reed-Solomon codes. Each of these rely on an irreducible 8th degree polynomial with ...
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2answers
213 views

Prove that there isn't a polynomial with $\text {f(x)}^{13} = {(x-1)}^{143}+(x+1)^{2002}$

Prove that there isn't a polynomial with $\text {f(x)}^{13} = {(x-1)}^{143}+(x+1)^{2002}$ We can easily find out that $\text {deg}(f) = 154$ Then?
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1answer
215 views

The unique root on (1,2)

For any given $n\geq 2$, let $x^n=\sum\limits_{k=0}^{n-1}x^{k}$ be the equation, prove: there is only one real root which in (1,2).
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1answer
105 views

Solve a cubic polynomial given that one root is four times a second root? [duplicate]

How exactly would you solve the equation: "Solve the equation $10x^3+23x^2+5x−2=0$ given that one root is four times a second root." How would you go about solving this? Any help would be greatly ...
2
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2answers
106 views

Solve a cubic polynomial (given one root is four times a second root)?

So, I've been stuck on a question for a long time now: "Solve the equation $10x^3 + 23x^2 + 5x - 2 = 0$ given that one root is four times a second root." How would you go about solving this? Any ...
10
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5answers
434 views

Reducibility of $x^{2n} + x^{2n-2} + \cdots + x^{2} + 1$

Just for fun I am experimenting with irreducibility of certain polynomials over the integers. Since $x^4+x^2+1=(x^2-x+1)(x^2+x+1)$, I thought perhaps $x^6+x^4+x^2+1$ is also reducible. Indeed: ...