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MATH / Consider the polynomials P(x)=x⁶−x⁵−x³−x²−x and Q(x)=x⁴−x³−x²−1. Given that z₁,z₂,z₃, and z₄ are the roots of Q(x)=0, find P(z₁)+P(z₂)+P(z₃)+P(z₄).

Problem

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problem

Consider the polynomials P(x)=x6−x5−x3−x2−xP(x) = x^6-x^5-x^3-x^2-x and Q(x)=x4−x3−x2−1Q(x)=x^4-x^3-x^2-1. Given that z1,z2,z3z_1, z_2, z_3, and z4z_4 are the roots of Q(x)=0Q(x)=0, find P(z1)+P(z2)+P(z3)+P(z4).P(z_1)+P(z_2)+P(z_3)+P(z_4).
Plain-text mathematical notation (without MathML)
Consider the polynomials P(x)=x⁶−x⁵−x³−x²−x and Q(x)=x⁴−x³−x²−1.  Given that z₁,z₂,z₃, and z₄ are the roots of Q(x)=0, find P(z₁)+P(z₂)+P(z₃)+P(z₄).
Original LaTeX notation
Consider the polynomials $P(x) = x^6-x^5-x^3-x^2-x$ and $Q(x)=x^4-x^3-x^2-1$.  Given that $z_1, z_2, z_3$, and $z_4$ are the roots of $Q(x)=0$, find $P(z_1)+P(z_2)+P(z_3)+P(z_4).$

level

Level 5

type

Intermediate Algebra

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Source and history

task_id
3644fb45-22f5-5219-b838-1e2dbad2a9d9
task_key
intermediate~5falgebra--test--3644fb45-22f5-5219-b838-1e2dbad2a9d9
task_revision_id
1
config / split
intermediate_algebra / test

Official source

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