1
vote
1answer
27 views

On the existence of polynomial roots

Assume $F$ is a field, and $f\in F[x]$ is polynomial. To see that $f$ has a root in some extension of $F$, without loss of generality we can assume $f$ is irreducible. Indeed any polynomial $f$ is ...
3
votes
2answers
71 views

Root of the polynomial $x(x-1)(x-2)\cdots(x-K)=C$

Is there an analytic way to obtain the highest root of the polynomial $x(x-1)(x-2)\cdots(x-K)=C$ in terms of $K$ and $C$? The integer $K \ll x$ and the constant $C$ are known. The other way to ask ...
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 ...
0
votes
1answer
67 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 ...
2
votes
2answers
33 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: ...
0
votes
2answers
199 views

Use a given zero to write P(x) as a product of linear and irreducible quadratic factors

The polynomial in question is: $x^4 - 8x^3 - 19x^2 + 288x - 612$ and the zero is $4 - i$. What I don't understand is how to go from the given zero to factorizing, especially as it's imaginary. ...
0
votes
1answer
35 views

$\frac{x^4 - x^3 + ax^2 + bx + c}{x^3 + 2x^2 - 3x + 1}$, remainder $3x^2 - 2x + 1$. Find $(a + b)c$.

Given the polynomials $P(x) = x^4 - x^3 + ax^2 + bx + c\\ Q(x) = x^3 + 2x^2 - 3x + 1\\ R(x) = 3x^2 - 2x + 1$ such that $P(x) = D(x)Q(x) + R(x)$, find $(a + b)c$. I would normally apply little ...
3
votes
1answer
238 views

Solve $x+y+z = x^3 + y^3 + z^3 = 8$ in $\mathbb{Z}$

Solve $x+y+z = x^3 + y^3 + z^3 = 8$ in $\mathbb{Z}$ First I tried to transform this equation, substituting $x = 8-y-z$. So I end up with: $$x^3 + y^3 + z^3 = 8$$ $$(8-y-z)^3 + y^3 + z^3 = 8$$ ...
3
votes
6answers
433 views

Solving $\sqrt{7x-4}-\sqrt{7x-5}=\sqrt{4x-1}-\sqrt{4x-2}$

Where do I start to solve a equation for x like the one below? $$\sqrt{7x-4}-\sqrt{7x-5}=\sqrt{4x-1}-\sqrt{4x-2}$$ After squaring it, it's too complicated; but there's nothing to factor or to ...
0
votes
1answer
27 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 + ...
3
votes
4answers
253 views

Solving this 3-degree polynomial

I'm trying to factor the following polynomial by hand: $-x^3 + 9x^2 - 24x + 20 = 0$ The simplest I could get is: $-x^2(x-9) - 4(5x+5) = 0$ Any ideas on how I could go ahead and solve this by hand? ...
18
votes
4answers
756 views

Find all roots of $\,(x + 1)(x + 2)(x + 3)^2(x + 4)(x + 5) = 360$

The question is to find all complex roots of $$(x + 1)(x + 2)(x + 3)^2(x + 4)(x + 5) = 360$$ and it is meant to be solved by hand. Is there any quick way to solve this using some trick that I'm not ...
0
votes
1answer
85 views

How to solve for x

$W(4d^2 (1-x^2)^2) = abc^3x \sqrt{(\pi^2 (i-x^2)^2 + 16 x^2) }$ I have to find x ,i have the values of all other constants , I tried to separate it using partial fraction but I am stuck. a=3 b=4 c=7 ...
1
vote
3answers
146 views

Adjunction of a root to a UFD

Let $R$ be a unique factorization domain which is a finitely generated $\Bbbk$-algebra for an algebraically closed field $\Bbbk$. For $x\in R\setminus\{0\}$, let $y$ be an $n$-th root of $x$. My ...