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Omni-MATH / Let r₁,r₂,r₃,r₄ be the four roots of the polynomial x⁴−4x³+8x²−7x+3. Find the value of (r₁²)/(r₂²+r₃²+r₄²)+(r₂²)/(r₁²+r₃²+r₄²)+(r₃²)/(r₁²+r₂²+r₄²)+(r₄²)/(r₁²+r₂²+r₃²)

Problem

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problem

Let r1,r2,r3,r4r_{1}, r_{2}, r_{3}, r_{4} be the four roots of the polynomial x4−4x3+8x2−7x+3x^{4}-4 x^{3}+8 x^{2}-7 x+3. Find the value of r12r22+r32+r42+r22r12+r32+r42+r32r12+r22+r42+r42r12+r22+r32\frac{r_{1}^{2}}{r_{2}^{2}+r_{3}^{2}+r_{4}^{2}}+\frac{r_{2}^{2}}{r_{1}^{2}+r_{3}^{2}+r_{4}^{2}}+\frac{r_{3}^{2}}{r_{1}^{2}+r_{2}^{2}+r_{4}^{2}}+\frac{r_{4}^{2}}{r_{1}^{2}+r_{2}^{2}+r_{3}^{2}}
Plain-text mathematical notation (without MathML)
Let r₁,r₂,r₃,r₄ be the four roots of the polynomial x⁴−4x³+8x²−7x+3. Find the value of (r₁²)/(r₂²+r₃²+r₄²)+(r₂²)/(r₁²+r₃²+r₄²)+(r₃²)/(r₁²+r₂²+r₄²)+(r₄²)/(r₁²+r₂²+r₃²)
Original LaTeX notation
Let $r_{1}, r_{2}, r_{3}, r_{4}$ be the four roots of the polynomial $x^{4}-4 x^{3}+8 x^{2}-7 x+3$. Find the value of $\frac{r_{1}^{2}}{r_{2}^{2}+r_{3}^{2}+r_{4}^{2}}+\frac{r_{2}^{2}}{r_{1}^{2}+r_{3}^{2}+r_{4}^{2}}+\frac{r_{3}^{2}}{r_{1}^{2}+r_{2}^{2}+r_{4}^{2}}+\frac{r_{4}^{2}}{r_{1}^{2}+r_{2}^{2}+r_{3}^{2}}$

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Source and history

task_id
d3c7136f-d69f-5ee3-9074-6b2981689734
task_key
test--d3c7136f-d69f-5ee3-9074-6b2981689734
task_revision_id
1
config / split
/ test

Official source

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