3
votes
2answers
52 views

Solutions for quartic

Suppose I have an equation in the form $(x-a)^4 + (x-b)^4 = c$. What is a clever way to find all four solutions? I have tried expanding and then used long division. However, I believe a better way is ...
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
2answers
37 views

How is the nature of the roots of a third degree polynomial determined?

Given a polynomial $p(x) = x^3-bx^2+cx-d = 0 $ such that all three roots are real positive integers. How does one figure out if the three roots are distinct? The coefficient of $x^3$ is 1. In the case ...
2
votes
5answers
126 views

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

How can we solve this equation? $x^4-8x^3+24x^2-32x+16=0.$
1
vote
1answer
71 views

Find the solutions of the equation…

How can I solve this equation? $$ \begin{equation*} \sqrt[3]{x-2}+\sqrt{x-1}=5 \end{equation*} $$ Frankly, I just have no idea at all!!! Thank you in advance!
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
2answers
81 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
2answers
36 views

Finding root for the segment - found the formula but it doesn't work for some values - wrong formula?

I have the segment, defined as $(x_1, y_1)$, $(x_2, y_2)$. I know that $y_1\ge 0$ and $y_2 < 0$. I want to compute the root point for that segment. I decided to do it that way: ...
0
votes
2answers
49 views

Square root of negative integer

Can I write: $-\sqrt{(2)}$ = $\sqrt{(-2)}$ and vice versa? Or, say, we have, $(-\sqrt{(x - 4)}$ Can this be changed into $(\sqrt{(4 - x)}$ by taking the minus sign inside the square root? How?
0
votes
2answers
82 views

Intriguing Equation

How many ordered tuples of 7 integers ${\{x_{i}\}}_{i=1}^{7}$ are there, such that $$\sum _{i=1}^{7}{x_{i}}-\prod_{i=1}^{7}{x_{i}} =6$$ where $1\le x_i\le 8$. I tried taking ${ \{ x_{ i }\} }_{ ...
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 ...
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.
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 ...
2
votes
1answer
91 views

Find the value of $m + n + r$

One of the roots of the equation $2000x^6+100x^5+10x^3+x-2=0$ is of the form $\frac{m+\sqrt{n}}r$ ,where $m$ is non zero integer and $n$ and $r$ are relatively prime numbers.Then the value of $m+n+r$ ...
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 ...
0
votes
4answers
134 views

How can I solve $y^{3}-3y^{2}+2=0$?

I am stuck at this equation $y^{3}-3y^{2}+2=0$. How do I solve it without calculator? It might be very trivial so I think I just need a hint. It is actually a substitution $y=\log x$,but I think it ...
1
vote
1answer
84 views

Rational Root theorem issue

I've given my class an example: $$2x^3+3x^2+6x+4=0$$ By the rational root theorem, if there is a rational root then it should be of the form $\frac{p}{q}$ where $p$ is a factor of 4 and $q$ is a ...
3
votes
3answers
170 views

$\alpha,\beta,\gamma$ are roots of cubic equation $x^3+4x-1=0$

If $\alpha,\beta,\gamma$ are the roots of the equation $x^3+4x-1=0$ and $\displaystyle \frac{1}{\alpha+1},\frac{1}{\beta+1},\frac{1}{\gamma+1}$ are the roots of the equation $\displaystyle ...
0
votes
2answers
113 views

If two polynomials both of n degree have n identical real roots, are they equal? Proof?

CORRECTION: The polynomials don't have to be equal, but one has to be a constant multiple of the other. I ask the question because I saw this fact used in this solution to a problem: Problem: Given ...
10
votes
4answers
163 views

Coefficients of a polynomial also are the roots of the polynomial?

How many real solutions $(r_1, r_2, \cdots, r_n)$ are there such that $(r_1, r_2, \cdots, r_n)$ are the roots of the polynomials $x^{n} + r_1 x^{n-1} + r_2 x^{n-2} + \cdots + r_n$ For $n = 2, 3, 4$ I ...
0
votes
2answers
60 views

A 3rd degree polynomial $P(x)$ has three unequal real roots. What is the least possible # of unequal real roots for $P(x^2)$

I got that if P(x) is a 3rd degree polynomial then P($x^2$) must be a 6th degree polynomial. I don't know how to proceed from here.
2
votes
4answers
115 views

Find the solution of the equation

Find all real solutions of this equation : $$x=\sqrt{2+\sqrt{2-\sqrt{2+x}}}$$
2
votes
2answers
31 views

Polynomial With Imaginary Roots

Working on question 1 here http://www.sosmath.com/cyberexam/precalc/EA2002/EA2002.html Find a polynomial with integer coefficients that has the following zeros: ...
17
votes
6answers
2k views

Can $x^3+3x^2+1=0$ be solved using high school methods?

I encountered the following problem in a high-school math text, which I wasn't able to solve using factorization/factor theorem: Solve $x^3+3x^2+1=0$ Am I missing something here, or is indeed a more ...
1
vote
1answer
83 views

Finding the solutions of $\cos (x) +x = a$

What is the approach to finding the solutions of the following function? I was not able to analytically resolve the solutions - but rather resorted to a graphical approach. $$\cos (x) + x = 1$$ or in ...
0
votes
2answers
137 views
1
vote
1answer
124 views

Is my simple (in my opinion) way of solving cubic equations correct?

I've been analyzing ways of solving cubic equations and I've come up with this one. I've tried to make it as simple as possible. So I'll show you a way of solving cubic equations when none of the ...
3
votes
1answer
1k views

Find all real zeros of $f(x)=2x^3+10x^2+5x-12$

Hey guys I'm having a little trouble with one problem: Find all real zeros of $$f(x)=2x^3+10x^2+5x-12.$$ I got $x=-4,(2x^2+2x-3)$. I'm just having trouble using the quadratic formula to get ...
2
votes
0answers
41 views

Is there a known algorithm for approximating all the real and imaginary zeros of any well behaved equation of a single variable?

Does there currently exist a general algorithm (or set of algorithms used together) that will approximate all the zeros of any well behaved non-differential equation of a single variable which has a ...
0
votes
5answers
320 views

Why doesn't $1/x=0$ have any solution?

Just out for curiosity ! Why $1/x=0$ doesn't have any solution? Or is it that the solution takes you to $1=0$ situation which would nullify mathematical principle that we stood for years Educate ...
-2
votes
2answers
90 views

Number of real roots ,eh!! [closed]

Consider $P(x)=1+x+\frac {x^2}{2}+\frac {x^3}{6}+\frac {x^4}{24}$ Find the number of real zeros of the polynomial $P(x)$
3
votes
2answers
46 views

Solve $\left(x^{2010}+1\right)\left(1+x^2+x^4+x^6+…+x^{2008}\right)=2010x^{2009}$

Solve for $x$ $\left(x^{2010}+1\right)\left(1+x^2+x^4+x^6+.......+x^{2008}\right)=2010x^{2009}$ solution should be by hand
6
votes
3answers
160 views

Solve $\lfloor{x}\rfloor$+$\lfloor2x\rfloor+\lfloor4x\rfloor+\lfloor16x\rfloor+\lfloor32x\rfloor=12345$

Solve for $x$ $$\lfloor{x}\rfloor+\lfloor2x\rfloor+\lfloor4x\rfloor+\lfloor16x\rfloor+\lfloor32x\rfloor=12345$$ I tried to put $x$=$I$+$f$ where $I$ is integer part and $f$ is fractional part but that ...
2
votes
4answers
147 views

Find number of solutions of $2^x$+$3^x$+$4^x$=$5^x$

Find number of solutions of $$2^x+3^x+4^x=5^x$$ I tried using graphs but don't know how to draw graph of L.H.S.
4
votes
4answers
108 views

Comment upon nature of the roots

How many roots are there of the following polynomial? How many are real, and how many are complex? ...
1
vote
1answer
27 views

Plot implicit equation

I'm working with a frequency-response curve of a nonlinear oscillator and came across the following equation (Kovacic & Brennan 2011, p. 179): $$ A^2 = \frac{f^2}{4 \xi^2 \omega^2 + (\omega^2 - ...
0
votes
3answers
211 views

$\log x =Cx^4$ has only one root. Find C

$\log x =Cx^4$ has only one root. Find C. I don't know how to solve this problem. Do you take derivative on both sides? I am thinking C equals 0. Am I correct on that?
1
vote
3answers
2k views

cubic equations which have exactly one real root

Question is to check : For any real number $c$, the polynomial $x^3+x+c$ has exactly one real root . the way in which i have proceeded is : let $a$ be one real root for $x^3+x+c$ i.e., we have ...
0
votes
3answers
443 views

Determine the number of real roots in the equation…

Determine the number of real roots in the equation $2x^3 + x^2 = 3$. I know about finding the different roots, and solving giving that it has (for example) 2i as a root, but I'm not sure how to just ...
2
votes
1answer
49 views

Solve the equation for $X$

$$X^3-3X^2+3X=\frac{3R-10}{2}$$ How can i solve it for $X$ ? I tried to do : $$\Rightarrow X(X^2-3X+3)=\frac{3R-10}{2}$$ ???
1
vote
3answers
334 views

Finding the zeros of $f(x)=-x^3-x^2+7x+7$

$$f(x)=-x^3-x^2+7x+7$$ it needs to be solved for the zeros I need to figure out the answer to this please help I have tried many different things and I'm confused
1
vote
2answers
47 views

Help for two values of x

I'm looking for help. Even if you just tell me the process rather than the answer. Given that $y=10-3x^2$, find two values of $x$ for which $y=-17$. How would I go about answering this?
3
votes
2answers
772 views

Finding the discriminant and roots of a polynomial

How is the discriminant of a polynomial determined? I know that for a quadratic function, the roots (where $f(x)=0$) are found by $$x=\frac{-b\pm\sqrt{\Delta}}{2a}$$ and here $\Delta$ is the ...
1
vote
0answers
52 views

Roots of A Non-linear Equation

I have the following non-linear equation $$b_1\left(\frac{1}{f_1^2}-\frac{1}{(f_1-a_1)^2}\right)=b_2\left(\frac{1}{f_2^2}-\frac{1}{(f_2-a_2)^2}\right)$$ where $$f_1+f_2=A(\ \mbox{constant})$$ When I ...
2
votes
0answers
208 views

Cubic roots and Cardano formula

On solving the cubic equations, applying Cardano formula yield complex results. I wanted to evaluate the exact roots (not numerical) but I ended up with complex numbers/nested radicals. To get rid ...
0
votes
0answers
21 views

Dependence on Parameters of the Solution of a Non-linear Equation

I have the following equation for the delay in a queue\begin{align} d(f)=\frac{c(1-f)^2}{2(1-a)}+\frac{\lambda b}{2f(f-a)}\end{align} where $0\le f\le 1;\quad c,a=\lambda\tau, \ b=\tau^2$ or ...
3
votes
1answer
219 views

how to find the roots of a cubic equation?

Given a formula $$x^3+ax^2+bx+c=0$$ how can I get the value of x without having an $i$ in my roots? Because Cardano's formula does have imaginary numbers if the discriminant is less than zero. My ...
11
votes
2answers
359 views

Number of real positive roots of a polynomial?

Consider the polynomial $$f(x)=x((1+x^n)^n+a^n)-a(1+x^n)^n,$$ where $n\geq 2$ is a positive integer and $a$ is a positive real number. I'm interesting in deducing the number of positive real roots ...
0
votes
3answers
82 views

Find the eigenvalues of the matrices.

The characteristic equations for the two matrices are: $x^3-8x-7=0$ and $x^3-6x^2+11x-6=0$ I know that in order to find the eigenvalues, I need to factor these two equations out. I'm just having a ...
0
votes
2answers
59 views

Root of an exponential equation

Let $0 \le a \le 1$ and $-\infty < b < \infty$. I am looking for a solution of the exponential equation. $$ a^x + abx = 0. $$ I guess closed form expression of the root in terms of $a$ and $b$ ...