0
votes
6answers
73 views

How to find the roots of $-x^3+3x^2-7x+5 = 0$?

I would like to understand how to go about solving something like this, not just get the solution but some kind of methodology (that hopefully makes as much intuitive sense as possible); I honestly ...
3
votes
1answer
47 views

Nice polynomial reducibility: $x^n+4$

Problem: Find all $n\in \mathbb{N}$ such that $f(x)=x^n+4$ is reducible in $\mathbb{Z}[x]$. It seems $n=4k$ is the only one (the factorization follows easily from Sophie Germain's identity in this ...
1
vote
3answers
55 views

Solve the following equation: $\frac{1}{x^2}+\frac{1}{(4-\sqrt{3}x)^2}=1$

Solve the following equation: $$\frac{1}{x^2}+\frac{1}{(4-\sqrt{3}x)^2}=1$$ I know it's from a Math Olympiad but I don't know which and I couldn't find it on the internet. Expanding everything ...
0
votes
3answers
52 views

Polynomial of degree 5 divisible by other polynomials.

I need some help with a problem. Find a polynomial $f(x)$ of degree $5$ such that both of these properties hold: $f(x)-1$ is divisible by $(x-1)^3$. $f(x)$ is divisible by $x^3$. I can't seem to ...
4
votes
4answers
221 views

Polynomial $f(x)$ degree problem.

Suppose the polynomial $f(x)$ is of degree $3$ and satisfies $f(3)=2$, $f(4)=4$, $f(5)=-3$, and $f(6)=8$. Determine the value of $f(0)$. How would I solve this problem? It seems quite complicated... ...
0
votes
2answers
42 views

Roots Of Monic Cubic

I'm currently preparing for the USA Mathematical Talent Search competition. I've been brushing up my proof-writing skills for several weeks now, but one area that I have not been formally taught about ...
3
votes
2answers
71 views

Polynomial division problem

Let $f(x) = x^{10}+5x^9-8x^8+7x^7-x^6-12x^5+4x^4-8x^3+12x^2-5x-5. $ Without using long division (which would be horribly nasty!), find the remainder when $f(x)$ is divided by $x^2-1$. I'm not sure ...
0
votes
1answer
23 views

Not understanding steps in Algebraic simplification

The simplification in question is that the expression goes from $(4-x)(6-x)(3-x)-8(3-x)=0$, to $(3-x)(8-x)(2-x)=0$ I don't understand how one goes from the first expression to the second. I ...
2
votes
3answers
164 views

How can this equality be established by elementary algebraic means?

Let $x \geq 1$. Then is it true that $2x^3 - 3x^2 + 2 \geq 1$? If so, how can I show this using only elementary ideas such as factorisation? Of course, I can demonstrate this using the methods of ...
1
vote
3answers
31 views

Factor Cyclic Polynomial

Factor $(a+b)(b+c)(c+a)+abc$. I know this is a cyclic polynomial, but I don't know how to solve problems like this. What should I do?
3
votes
2answers
48 views

Find the value of $\frac{S_{5}S_{2}}{S_{7}}$

If $a$, $b$, $c$ $\in \mathbb R$, we define $S_{k}=\frac{a^k+b^k+c^k}{k}$ (where $k$ is a non-negative integer). Given that $S_{1}=0$, find the value of $$\frac{S_{5}S_{2}}{S_{7}}$$ I tried: ...
0
votes
3answers
43 views

Solving the complex polynomial

For the complex polynomial $z^3 -5z^2 +(7-2i)z +6i-3 = 0 $ $1)$ show that $2+i $ is a root. $2)$ solve the given equation. Attemp to solve: I'm not really sure how to solve this, but I ...
0
votes
2answers
56 views

give a complete factored form of the polynomial $-6a^5+48a^4+12a$

Give a complete factored form of the polynomial $-6a^5+48a^4+12a$ I have tried solving this equation and I just cant figure it out. Help me, and give me the answer.
0
votes
1answer
22 views

Relationship between constant term and roots

Does anyone know of a relationship between the constant term of a polynomial and the roots of the polynomial? Specifically, if we know the constant term, is it possible for a root which divides the ...
1
vote
2answers
58 views

How to find a polynomial with $f(1), f(4),f(9)$ prime and coefficients in $\{1,2,3…10\}$?

How to find a polynomial with $f(1), f(4),f(9)$ prime and coefficients in $\{1,2,3...10\}$? I can't think of any way on how to generate such types of polynomials? Also, would they have a minimum ...
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
33 views

AMM Polynomial equation

Solve the equation: $x^7+7px^5+14p^2x^3+7p^3x+q=0$ I've tried obvious things like factorization or maybe guessing a solution. I'd appreciate a solution not too far from high school level.
7
votes
1answer
80 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 ...
0
votes
6answers
68 views

solving the inequalty

are there any ways to solve :$ x^4 -6x^3 +28x^2 -64x +96 >0$ ?
4
votes
0answers
70 views

Irreducibility of some polynomial

Let $p(x) = (1+ \cdots +x^k)^2 + (1+ \cdots +x^k) + 1$, for some $k \geq 2$ fixed. I would like to know if $p(x)$ is irreducible in $\mathbb{Q}[x]$.
2
votes
0answers
39 views

How “separable” (not in that sense) is a polynomial?

Since "separable" is used for different meaning in separable polynomial and separation of variable, I am having trouble searching for anything related to my question. So I hope someone can help with ...
1
vote
2answers
86 views

Find the roots of the equation $(1+xi)^n+(1-xi)^n=0$

Find the roots of the equation $f(x)=(1+xi)^n+(1-xi)^n=0$. I'm having problems finding the roots...this is what I've done: First I expressed $(1+xi)^n$ and $(1-xi)^n$ in trigonometric form and ...
2
votes
1answer
54 views

Given roots (real and complex), find the polynomial

This is not a duplicate of theory of equations finding roots from given polynomial. Given that the roots (both real and complex) of a polynomial are $\frac{2}{3}$, $-1$, $3+\sqrt2i$, and $3+\sqrt2i$, ...
0
votes
5answers
69 views

Graphing polynomials

Sketch a graph of the polynomial $P(x)=(x-2)^2(x+1)^3$. You must plot and label the x and y intercepts and these should be the only points you plot. How do I sketch the graph of a polynomial?
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
3answers
71 views

Roots of $x^2 +2x +2$ Over $\mathbb{C}$

Find the roots of $x^2 +2x +2$ over $\mathbb{C}$ I need to prove somehow that the roots will be $(1 + i) , (1 - i)$ Any ideas how can I find those roots in a simple way?
1
vote
2answers
32 views

Using Remainder or Factor Theorems to Find Coefficient

I'm studying the remainder and factor theorems and a question asks: -4 is a root of $x^4 + ax^3 - 19x^2 - 46x + 120$ What is the value of a? Since -4 is a root then I can deduce that x+4 is a ...
0
votes
3answers
40 views

Explanation on characterstic polynomial

$A_2 = \begin{pmatrix} 1 & 1 \\ a & 1 \end{pmatrix} $ So the characteristic polynomial of $A_2$ is $P_a(t) = (t-1)^2 - a $ Then, $ P_a(t) = t^2 -2t +1 -a$ ...
2
votes
2answers
407 views

How can I find the roots of a quartic equation, knowing one of its roots?

I need to decompose (in $\Bbb{C}[x]$) the function $$ f(x) = x^4 + 4x^3 - 4x^2 + 24x + 15 $$ in its simplest form, knowing that $1 - 2i$ is one of its roots. Any ideas?
1
vote
3answers
41 views

Find the intersections of the functions

I have $f(x)=-x^2+4$ a parabola and $g(x)=\sqrt{(4-x^2})$ a semi circle with a raduis of $2$ if I say $g(x)=f(x)$ and solve for $x$. I should find the points at which $x$ intercepts ...
0
votes
2answers
25 views

Polynomial Divison of unknowns

How to complete this $$P(X)=\frac{2x^4-7x^3+5x^2+ax+b }{ 2x^2+x-1}$$ so that the division is without a remainder? When it is divided it gives two equation.
2
votes
2answers
72 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 ...
0
votes
1answer
57 views

Understanding 2012 AMC 12B #23

Monic quadratic polynomial $P(x)$ and $Q(x)$ have the property that $P(Q(x))$ has zeros at $x=-23$, $-21$, $-17$, and $-15$, and $Q(P(x))$ has zeros at $x=-59$,$-57$,$-51$ and $-49$. What 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 ...
1
vote
2answers
27 views

Changing a binomial denominator into a mixed expression?

Why does $\dfrac{k^3-1}{k-1}$ turn into $k^3 + 1$ when I'm dividing? The result of that would be $2$ which is the correct answer $k^2 - k + 1 + \dfrac{2}{k - 1}$. I only see myself getting $k^3 - ...
2
votes
3answers
36 views

A certain polynomial P(x) , $x\in R$ when divided by $x-a, x-b,x-c$ leaves the remainders a,b,c respectively…

A certain polynomial P(x) , $x\in R$ when divided by $x-a, x-b,x-c$ leaves the remainders a,b,c respectively. Find the remainder when P(x) is divided by $(x-a)(x-b)(x-c)$ is (a,b,c are distinct) My ...
0
votes
1answer
25 views

Factoring Polynomial Questions

How do you decide whether to use synthetic division or the factor theorem to help you factor a polynomial? Please help me answer.
0
votes
1answer
17 views

Relation between polynomials.

Let $P_{\lambda}(X) = X^4 + 6X^2 + \lambda X -3$ be a polynomial for every $\lambda \in\mathbb{C}$. Prove that if $\alpha \in \mathbb{C}$ is a root of multiplicity $2$ of $P$, then it is a root of ...
6
votes
3answers
80 views

Solving Equation of Degree n, where n is any value between 1 and 2

How does one solve an equation of the form: $$ax^n + bx + c = 0$$ where n is a non integer value between 1 and 2. Is there a formula to provide an analytic solution?
4
votes
2answers
38 views

The relationship between the intercepts and the remainder in the remainder theorem

The polynomial remainder theorem states that when a polynomial $P(x)$ of degree $> 0$ is divided by $x-r$ ($r$ being some constant) the remainder is equal to $P(r)$, that is: $$\begin{array}l If ...
0
votes
2answers
29 views

Show that $p+q^{2}=1$ where $x^{3}+px+q=0$ and one of the roots is the reciprocal of the other?

let the three roots be $z, 1/z, t$. So $z+1/z+t=0$ and $zt+1+t/z=p$ and $z(1/z)t=-q=t$ $-1/z-z=t$ $p+q^{2}=zt+1+t/z + t^{2}$ How do I simplify the RHS to get 1?
6
votes
4answers
113 views

Solve $x^{3}-3x=\sqrt{x+2}$

Solve for real $x$ $$x^{3}-3x=\sqrt{x+2}$$ By inspection, $x=2$ is a root of this equation. So, I squared both sides and divided the six degree polynomial obtained by $x-2$. Then I got a ...
0
votes
1answer
19 views

$\gcd(f,f')=1$ Does this imply that f has not multiply irreducible factors in $\mathbb{C}[x]$?

I want to find out if this affermation is true: let $f\in \mathbb{Q}[x]$ such that $\gcd(f,f')=1$ Does this imply that f has not multiply irreducible factors in $\mathbb{C}[x]$? (We know that it has ...
2
votes
3answers
93 views

How does $\frac{t^2}{t+1}$ equal $t-1+\frac{1}{t+1}$?

I do the long division: 1: t+1 goes into $t^2$ t times 2: Subtract $t^2$ + 1 from $t^2$ and get -1 3: Answer: t - $\frac{1}{t+1}$ Am I missing something here?
5
votes
1answer
84 views

Find $\lfloor {\alpha}^6 \rfloor$

If $\alpha$ is a real root of the equation $$x^5-x^3+x-2=0$$ find the value of $\lfloor {\alpha}^6 \rfloor$. This one totally stumped me. We are asked to calculate $\lfloor {\alpha}^6 ...
7
votes
3answers
114 views

How to solve the following? $ x^3+1=2{(2x-1)}^{1/3} $.

Find all the real solutions of $$x^3+1=2{(2x-1)}^{1/3} $$ I tried to cube both sides but got messed up with a nine degree equation! Please help. Thanks in advance!
3
votes
1answer
54 views

Proving no polynomial $P(x)$ exists such that $P(a) = b$, $P(b) = c$, $P(c) = a$

If $P(x)$ is a polynomial with integer coefficients and $a, b ,c$ are three distinct integers, then show that it is impossible to have $P(a) = b$, $P(b) = c$, $P(c) = a$.
1
vote
2answers
43 views

Remainder theorem question

If $n$ is an integer, what is the remainder when $$3x^{2n+3}-4x^{2n+2}+5x^{2n+1}-8$$ is divided by $x+1$? How would we know what the value of $n$ is?
0
votes
1answer
64 views

Prove that if polynomial $f(x) = x^6 + ax^3 + bx^2 + cx + d$ null places are real, then$ a = b = c = d = 0$

if polynomial null places are all real, then how can i show that $a = b = c = d = 0$? $x(x(0x + 0 + x^4) + 0) + 0 => x^6 = f(x)$ ? But that doesn't prove anything for me i think
5
votes
0answers
60 views

Minimising an expression - involving polynomial

I found this one on a forum but it has been unanswered from long there. I am curious to know if there is a solution to this problem. Here it is: Let n be a positive integer. Determine the smallest ...