0
votes
2answers
26 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
67 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
162 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?
2
votes
2answers
45 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
20 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
32 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
64 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
404 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
24 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
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 ...
0
votes
1answer
54 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
184 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
35 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
3answers
89 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 ...
4
votes
1answer
80 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 ...
6
votes
3answers
112 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
42 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
62 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
58 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 ...
0
votes
3answers
78 views

Find all values of $a$ for which there are two real solutions of $x^3-2ax^2+a^2x-3=0$

Find all values of $a$ for which there are two real solutions of the equation. $$x^3-2ax^2+a^2x-3=0$$ Ans = $1.5\sqrt[3]{6}$ I tried to research the function by dint of derivative, but it didn't ...
5
votes
4answers
96 views

What is the minimum value of $abc$

If the roots of the equation $$ax^2-bx+c=0$$ lie in the interval $(0,1)$, find the minimum possible value of $abc$. Edit: I forgot to mention in the question that $a$, $b$, and $c$ are natural ...
0
votes
3answers
59 views

How does knowing the degree of a polynomial help us?

Any polynomial such as $5x^6+2x^3+8$ has the degree of polynomial 6 but not 3 as it is said that the highest power of any term in a polynomial is the degree of that polynomial. But why? And how ...
4
votes
2answers
64 views

Evaluate $a+b+c+d$

If $a$, $b$, $c$, and $d$ are distinct integers such that $$(x-a)(x-b)(x-c)(x-d)=4$$ has an integral root $r$, what is the value of $a+b+c+d$ in terms of $r$? I tried to analyze graphically by ...
3
votes
2answers
58 views

Find the value of $\left | b-c \right |$

Given that $a, b, c \in \mathbb{Z}$, $a>10$ and $$(x-a)(x-12)+2=(x+b)(x+c)$$ Find the value of $\left | b-c \right |$ NOTE: The answer to this problem (as given on the last page of my book) is ...
0
votes
1answer
31 views

Integer polynomial with two aligned roots

Anyone have an example of a monic polynomial $f(x) \in \mathbb{Z}[x]$ with $f(0)=1$ such that for $\alpha \in \mathbb{C}\backslash\mathbb{R}$ and $t \in R, t>0, t\neq 1$ both $\alpha$ and $t\alpha$ ...