If p, q, r are the roots of $x^3-6x^2+3x+1=0$ determine the possible values of $p^2q+q^2r+pr^2$. I tried to solve this problem through Vieta's relations but I did not find a way that allows not to use the cubic formula. I found this question on Pathfinder of Olympiad Mathematics and I have been struggling for a week without getting anywhere.
2026-04-01 15:10:10.1775056210
If p, q, r are the roots of $x^3-6x^2+3x+1=0$ determine the possible values of $p^2q+q^2r+pr^2$.
696 Views Asked by Bumbble Comm https://math.techqa.club/user/bumbble-comm/detail At
1
There are 1 best solutions below
Related Questions in ALGEBRA-PRECALCULUS
- How to show that $k < m_1+2$?
- What are the functions satisfying $f\left(2\sum_{i=0}^{\infty}\frac{a_i}{3^i}\right)=\sum_{i=0}^{\infty}\frac{a_i}{2^i}$
- Finding the value of cot 142.5°
- Why is the following $\frac{3^n}{3^{n+1}}$ equal to $\frac{1}{3}$?
- Extracting the S from formula
- Using trigonometric identities to simply the following expression $\tan\frac{\pi}{5} + 2\tan\frac{2\pi}{5}+ 4\cot\frac{4\pi}{5}=\cot\frac{\pi}{5}$
- Solving an equation involving binomial coefficients
- Is division inherently the last operation when using fraction notation or is the order of operation always PEMDAS?
- How is $\frac{\left(2\left(n+1\right)\right)!}{\left(n+1\right)!}\cdot \frac{n!}{\left(2n\right)!}$ simplified like that?
- How to solve algebraic equation
Related Questions in POLYNOMIALS
- Alternate basis for a subspace of $\mathcal P_3(\mathbb R)$?
- Integral Domain and Degree of Polynomials in $R[X]$
- Can $P^3 - Q^2$ have degree 1?
- System of equations with different exponents
- Can we find integers $x$ and $y$ such that $f,g,h$ are strictely positive integers
- Dividing a polynomial
- polynomial remainder theorem proof, is it legit?
- Polyomial function over ring GF(3)
- If $P$ is a prime ideal of $R[x;\delta]$ such as $P\cap R=\{0\}$, is $P(Q[x;\delta])$ also prime?
- $x^{2}(x−1)^{2}(x^2+1)+y^2$ is irreducible over $\mathbb{C}[x,y].$
Related Questions in ROOTS
- How to solve the exponential equation $e^{a+bx}+e^{c+dx}=1$?
- Roots of a complex equation
- Do Irrational Conjugates always come in pairs?
- For $f \in \mathbb{Z}[x]$ , $\deg(\gcd_{\mathbb{Z}_q}(f, x^p - 1)) \geq \deg(\gcd_{\mathbb{Q}}(f, x^p - 1))$
- The Heegner Polynomials
- Roots of a polynomial : finding the sum of the squares of the product of two roots
- Looking for references about a graphical representation of the set of roots of polynomials depending on a parameter
- Approximating the first +ve root of $\tan(\lambda)= \frac{a\lambda+b}{\lambda^2-ab}$, $\lambda\in(0,\pi/2)$
- Find suitable scaling exponent for characteristic polynomial and its largest root
- Form an equation whose roots are $(a-b)^2,(b-c)^2,(c-a)^2.$
Related Questions in CUBICS
- Roots of a complex equation
- Cubic surfaces and 27 lines
- Polynomial Equation Problem with Complex Roots
- Cubic Discriminant
- Is it always possible to rearrange an equation desirably?
- Form an equation whose roots are $(a-b)^2,(b-c)^2,(c-a)^2.$
- if $x^3 + px^2+qx+r = 0$ has three real roots, show that $p^2 \ge 3q$
- The complex equation $x^3 = 9 + 46i$ has a solution of the form $a + bi$ where $a,b\in \mathbb Z$. Find the value of $a^3 + b^3$
- Roots of $z^3 + 3iz^2 + 3z + i = 0$?
- If the roots of the cubic equation $ax^3+bx^2+cx+d=0$ are equal, can one then establish a relationship between $a, b, c, d$?
Trending Questions
- Induction on the number of equations
- How to convince a math teacher of this simple and obvious fact?
- Find $E[XY|Y+Z=1 ]$
- Refuting the Anti-Cantor Cranks
- What are imaginary numbers?
- Determine the adjoint of $\tilde Q(x)$ for $\tilde Q(x)u:=(Qu)(x)$ where $Q:U→L^2(Ω,ℝ^d$ is a Hilbert-Schmidt operator and $U$ is a Hilbert space
- Why does this innovative method of subtraction from a third grader always work?
- How do we know that the number $1$ is not equal to the number $-1$?
- What are the Implications of having VΩ as a model for a theory?
- Defining a Galois Field based on primitive element versus polynomial?
- Can't find the relationship between two columns of numbers. Please Help
- Is computer science a branch of mathematics?
- Is there a bijection of $\mathbb{R}^n$ with itself such that the forward map is connected but the inverse is not?
- Identification of a quadrilateral as a trapezoid, rectangle, or square
- Generator of inertia group in function field extension
Popular # Hahtags
second-order-logic
numerical-methods
puzzle
logic
probability
number-theory
winding-number
real-analysis
integration
calculus
complex-analysis
sequences-and-series
proof-writing
set-theory
functions
homotopy-theory
elementary-number-theory
ordinary-differential-equations
circles
derivatives
game-theory
definite-integrals
elementary-set-theory
limits
multivariable-calculus
geometry
algebraic-number-theory
proof-verification
partial-derivative
algebra-precalculus
Popular Questions
- What is the integral of 1/x?
- How many squares actually ARE in this picture? Is this a trick question with no right answer?
- Is a matrix multiplied with its transpose something special?
- What is the difference between independent and mutually exclusive events?
- Visually stunning math concepts which are easy to explain
- taylor series of $\ln(1+x)$?
- How to tell if a set of vectors spans a space?
- Calculus question taking derivative to find horizontal tangent line
- How to determine if a function is one-to-one?
- Determine if vectors are linearly independent
- What does it mean to have a determinant equal to zero?
- Is this Batman equation for real?
- How to find perpendicular vector to another vector?
- How to find mean and median from histogram
- How many sides does a circle have?
Hint: Let \begin{eqnarray*} A=p^2q+q^2r+r^2p \\ B=pq^2+qr^2+rp^2. \end{eqnarray*} Now calculate $A+B$ and $AB$ ... and solve the quadratic.