0
votes
0answers
24 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. ...
0
votes
0answers
38 views

Number of roots of a polynomial (Proof)

What might be a simple proof to show that the maximum number of roots of a polynomial is equal to the degree of the polynomial? For example a quadratic polynomial can have a maximum of 2 roots. Can ...
0
votes
4answers
78 views

Why all such polynomials have $-1$ as a root?

Why all polynomials of this form have $-1$ as a root? $ x^5+x^4+x^3+x^2+x+1 $ and similar polynomials like $ x^9+x^8+x^7+x^6+x^5+x^4+x^3+x^2+x+1$
1
vote
1answer
44 views

Number of integer roots possible of the following polynomial

Let $p(x)$ be polynomial with integer coefficients, such that $p(0)$ and $p(1)$ are both odd. What is the maximum possible number of integer roots this polynomial can have?
5
votes
1answer
80 views

Prove that the equation $1+x+\dfrac{x^2}{2!}+\cdots+\dfrac{x^n}{n!}$ cannot have a multiple root.

Prove that the equation $$1+x+\dfrac{x^2}{2!}+\cdots+\dfrac{x^n}{n!}$$cannot have a multiple root. Using induction and the result that $f(x)=0$ have a root $\alpha$ of multiplicity $r\implies ...
1
vote
0answers
25 views

Find all integers $m$ and positive integers $n > 1$ so that $m + \sum_{k=1}^n x^k/k!$ has a rational root

If $m = 1$, then $m + \sum_{k=1}^n x^k/k!$ has no rational root for $n > 1$. And clearly the polynomial has a rational foot for all integers $m$ if $n = 1$. So, besides those cases, for what ...
0
votes
1answer
20 views

Show that a Polynomial has certain factorization

$P(x)$ is a polynomial in $x$ of degree $\leq n-1$. Show that $P(x)$ has $n-1$ distinct roots and thus has the factorization $$k\Pi_{i=2}^n(x-a_i)$$, where the constant $k$ is the coefficient of ...
4
votes
2answers
78 views

If $a,b,c(a,b,c\in\mathbb{R} )$ satisfy $b^2-4ac<0$ then equation $f(x)=0$ has complex root

I would appreciate if somebody could help me with the following problem: Q: show that ($n>2, n\in\mathbb{N}$) Let $f(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots+ax^2+bx+c, ...
2
votes
5answers
127 views

How to solve $x^4-8x^3+24x^2-32x+16=0$

How can we solve this equation? $x^4-8x^3+24x^2-32x+16=0.$
3
votes
4answers
167 views

Find all roots of $x^{6} + 1$

I'm studying for my linear algebra exam and I came across this exercise that I can't solve. Find all roots of polynomial $x^{6} + 1$. Hint: use De Moivre's formula. I guessed that two roots are $i$ ...
5
votes
1answer
76 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 ...
4
votes
0answers
78 views

All roots of a polynomial lie on a circle.

I'm stuck in the following problem and I need your help to solve it. Given a number $\alpha$, $0 < \alpha < 1$. $A_j(x)$ is a sequence of polynomials of $x^{-1}$ such that: $A_0(x) = 1; \\ ...
4
votes
6answers
157 views

$f(x)=x^3+ax^2+bx+c$ where $1\ge a\ge b\ge c\ge 0$. If $\lambda$ is any root of the polynomial, show that $|\lambda|\le 1$

$f(x)=x^3+ax^2+bx+c$ where $1\ge a\ge b\ge c\ge 0$. If $\lambda$ is any root of the polynomial, show that $|\lambda|\le 1$. My attempt: As the polynomial is a cubic, it must have atleast one real ...
1
vote
1answer
56 views

Working out the discriminant to a polynomial and using for working out “a”

For an equation: $$ x-b^2/x^3+a=0 \\$$ i.e. $$ x^4-b^2+ax^3=0 \\$$ If the discriminant is positive (i.e. $> or =0$) for real roots, what is the discriminant for these equations? Can you use the ...
4
votes
2answers
91 views

Can you find a Polynomial of Degree 7 that has 2 complex roots and 5 real?

Can you find a Polynomial of Degree 7 that has 2 complex roots and 5 real? The polynomial, call it $f(x)$ must be irreducible over $\mathbb{Q}$ (or over $\mathbb{Z}$ as Gauss' lemma can be used.) ...
1
vote
0answers
78 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 ...
0
votes
0answers
17 views

The number of roots of the system of equations in finite field

Let q be a prime power, $GF(q)$ be a finite field and $GF(q)[x]$ the polynomial ring over GF(q). For $m \in \mathbb{Z}_{>0}$: $$f_1(x),f_2(x),\dots,f_m(x) \in GF(q)[x]$$ and each degree is at ...
0
votes
2answers
82 views

How to solve for a non-factorable cubic equation?

I want to know how one would go about solving an unfactorable cubic. I know how to factor cubics to solve them, but I do not know what to do if I cannot factor it. For example, if I have to solve for ...
1
vote
4answers
55 views

Let $f(x)=x^2+17x+a$, $g(x)=x^2-17x-a$, $r$ a root of $f$ and $-r$ a root of $g$. Determine the roots of $f$.

Let $f(x)=x^2+17x+a$ and $g(x)=x^2-17x-a$. Suppose $r$ is a root of $f$ and $-r$ is a root of $g$. Determine all roots of $f$. From the descriptions, I can conclude that $f(x)-g(x)=2a$. But that ...
1
vote
1answer
47 views

Solving multivariate polynomial to find closest point to a $3$ (or more) circles

My requirement is to find the point closest to three circles. So lets say the three circles are $C_1$, $C_2$, $C_3$. I want to find the point in the space such that the SUM of its distance from $C_1$, ...
0
votes
1answer
56 views

An Application of Rouche's Theorem to Two Cases

Here is my question - it is an example sheet question, completely non-examinable: [I have managed this first part, but am including it to help give a sense of where the question is going.] $(i)$ ...
1
vote
1answer
94 views

Solution of cubic modulo some prime

Let $f(x)=x^3+3x+12$. Now if we have the relation $$f(x)\equiv0\pmod p$$ for some prime $p$, then what are the values of $p$ for which this equation is solvable for $x$? I know that the cubic ...
1
vote
3answers
311 views

Polynomials with Integer Coefficients and irrational roots

Is there a polynomial with integer coefficients which has √2 +√7  as a root?
0
votes
0answers
15 views

Find roots of characteristic equation $p(a,k)=0$

I need help understanding a derivation I've seen in a paper. I have a characteristic equation expressed as a polynomial $p(a,k)$, which means I can represent the characteristic equation as $p_a(a,k) ...
1
vote
2answers
35 views

Separability of $f(x) = (x-1)^2(x-3)$ over $\mathbb{Q}$

This is an example in Ash, Basic Abstract Algebra, ch.3.4 page 73 at the bottom (or here on page 11). It states that $f(x) = (x-1)^2(x-3)$ over $\mathbb{Q}$ is separable. But, $f'(x) = ...
0
votes
0answers
35 views

Finding roots and studying the sign if a polynomial?

We have two polynomials $g(x):= 1+x+\cdots+x^{2m+1}$ and $f(x):= 1+x+\frac{x²}{2}+\cdots+\frac{x^n}{n}$. For the first one, we wish to find the real roots and study the sign as $x$ varies. I ...
2
votes
0answers
37 views

Showing that the n first derivatives of (x²-1)^n have at least r roots (for the r-th derivative)?

I have f(x) = (x²-1)^n. I want to show that, for r = 0,1,2,...,n, the r-th derivative is a polynomial (that's easy to show) that has no fewer than r distinct roots in (-1,1). I guess I need to use ...
2
votes
0answers
58 views

Descartes Rule of Sign for exponential sums

I have the following exponential sums ($x\in\mathbb{R}$) $$f(x)=\sum_{i=1}^Na_iP_i(x)b_i^x$$ where $P(x)$ is some monomial, e.g., $x^2, x^3,\dots$, so $f(x)$ looks like ...
2
votes
2answers
137 views

Solve for p in $\frac1{20} = (1 - p)^{19}p$

I need help to solve for $p$, where $p$ is a probability, i.e. is between $(0,1)$. $\frac1{20} = (1 - p)^{19}p$ How would one solve for $p$? Thnx
1
vote
1answer
38 views

Find coefficients so that polynomial has at least one rational root

I have the following problem: Given $P(X) = X^5 + 15aX^4 + 12bX^3 -18X^2 -1$ Find $a,b \in \Bbb Z$ so that $p$ has at least one rational root. Prove that for any $a, b$ the ...
1
vote
1answer
30 views

Finding a polynomial with product and sum of its zeroes

A was reading a book with this question in it: Q. Find a quadratic polynomial, the sum of whose zeroes is 7 and their product is 12. Hence find the zeores of the polynomial. Sol. Let the ...
1
vote
3answers
45 views

Relation betwen coefficients and roots of a polynomial, K.A.Stroud

I am stuck on example 3, page 4 of Advanced Engineering Mathematics. The equation to be solved is $x^3+3x^2-6x-8=0$, The solution gives the roots as $-4, 2,-1$. Is it possible for someone to show me ...
5
votes
2answers
350 views

Polynomial $p(a) = 1$, why does it have at most 2 integer roots?

The question that I am trying to answer is : Suppose is $p(x)$ is a polynomial with integer coefficients. Show that if $p(a) = 1$ for some integer a then $p(x)$ has at most two integer roots. I have ...
2
votes
0answers
46 views

Trigonometric functions of angle fractions

I've just encountered a problem that seems to me interesting enough so that some result exists on the subject. I was working on a problem in complex analysis, in which I needed the fifth root of a ...
2
votes
1answer
64 views

How many quadratic polynomials exist given the two zeroes? ($1$ or $\infty$)

I was reading some book which had this question: Q. The number of [quadratic] polynomials having zeros $-2$ and $5$ is: (A) 1 (B) 2 (C) 3 (D) More than three? Sol. (A) 1. But ...
2
votes
1answer
80 views

Solve exponential-polynomial equation

Solve the equation in $\mathbb{R}$ $$10^{-3}x^{\log_{10}x} + x(\log_{10}^2x - 2\log_{10} x) = x^2 + 3x$$ To be fair I wasn't able to make any progress. I tried using substitution for the ...
2
votes
2answers
44 views

Determine a quartic equation

I am working on a puzzle from Popular Science from the 1980's. This was a puzzle that existed before pocket calculators or programmable computers. It results in one needing to solve a seeming quartic ...
0
votes
3answers
84 views

$P(x)=x^5+ax^4+bx^3+cx^2+dx+e$ has roots $1,2,3,4$ and $k$. Find $P(5) -P(0)$.

A polynomial $P(x)$ with leading coefficient $1$ is of degree $5$, and its distinct roots are $1, 2, 3, 4$ and $k$. Find the value of $P(5) -P(0)$. I have no clue on what my initial steps should be.
4
votes
4answers
88 views

Why is the other root negative even though the coefficients are rational?

About quadratic equations, I have the follow question, What is the equation, with rational coefficients, knowing that one root is X1 = 1 + sqrt(3) ? So, to solve the problem, I must know that ...
0
votes
1answer
45 views

Four real roots of $x^4+2x^3+mx^2+2x+1=0$ iff $m$ is…

The Equation $x^4+2x^3+mx^2+2x+1=0$ has $4$ different real roots iff: a) $m<3$; b) $m<2$; c) $m<-6$; d) $1<m<3$; e) $-6<m<2$
2
votes
5answers
360 views

proving zeros of a polynomial are not real

I'm working on a optimization problem and need to show that \begin{equation} \frac{1}{2}x^4 - x^3 -x + 100 = 0 \end{equation} has no real solution in order to prove certain properties about the ...
2
votes
1answer
76 views

Game of polynomials

Written on a blackboard is the polynomial $x^2+x+2014$.Calvin and Peter take turns alternatively (starting with Calvin) in the following game. During his turn, Calvin should either increase or ...
1
vote
1answer
44 views

polynomial of degree 3 over set of rationals having only two rational zeros.

Does there exist a polynomial of degree 3 over the set of rationals which has only two rational zeros?
3
votes
1answer
72 views

Why Rational Root Theorem only works with integers

Why does the rational root theorem only work when the polynomial has integer coefficients?
5
votes
4answers
277 views

Understanding non-solvable algebraic numbers

Background We know from Galois theory that the zeros of a polynomial with rational coefficients whose Galois group is solvable can be expressed in a formula that involves rational powers of the ...
2
votes
0answers
48 views

Rational Non-Integral Root

Prove by contradiction that the following equation with integral coefficients can not have a rational but non integral root. $x^{n}+p_{n-1}x^{n-1}+p_{n-2}x^{n-2}+\cdots +p_{0}=0$
1
vote
2answers
98 views

Show quartic polynomial has no real solutions

To show a lower bound for the runtime of an algorithm, I need to show that $$ 3 x^4 - \frac{64}{5} x^3 + \frac{192}{5} x^2 - \frac{192}{5} x+ 12 > 0 $$ for all real numbers $x\in \mathbb{R}$. ...
0
votes
1answer
70 views

Is this polynomial solvable by radicals?

The polynomial $p(x) = x^6-9x^4-4x^3+27x^2-36x-23$. has at least one (real, irrational) root that is expressible by radicals (can you find it?). Are all the roots of $p$ expressible by radicals and ...
0
votes
1answer
56 views

Substitution to linear + nth power form

Given an arbitrary polynomial: $$a_0 + a_1x + a_2x^2 ... a_nx^n$$ Does there exist a series of substitutions (or single substitution if you choose to combine them) that leaves this function in the ...
3
votes
1answer
138 views

Proof Verification: The polynomial $f(x) = (x+1)^n-x^n-1$ has a root of multiplicity 2 if and only if $n \equiv 1 \pmod 6$

Proof Verification: The polynomial $f(x) = (x+1)^n-x^n-1$ has a root of multiplicity 2 if and only if $n \equiv 1 \mod 6$ Let $r$ be a root, real or complex, of multiplicity 2 of $f(x)$. Then, by the ...