Maths WorkshopMaths problems with full solutions

Polynomials

50 problems · Roots of polynomials, factorisation, divisibility and Vieta's formulas.

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Ex.261Degree of sums and products

#degree #operations #polynomial

Ex.333Remainder of a polynomial

#polynomials #euclidean division

Ex.336Sum of cubes of the roots

#vieta #polynomials

Ex.536Divisibility by $X-a$

#factor theorem #divisibility #roots

Ex.537Euclidean division

#Euclidean division #quotient #remainder

Ex.538The factor theorem

#factor theorem #roots #divisibility

Ex.539Horner's scheme

#Horner scheme #evaluation #algorithm

Ex.540Vieta's formulas for a quadratic

#Vieta #sum and product of roots #quadratic

Ex.598√2 + √3 is irrational

#rational root theorem #irrationality #minimal polynomial

Ex.599Power sums from the first three

#Vieta #Newton sums #symmetric polynomials

Ex.600The remainder of a huge power

#polynomial division #remainder #roots of unity

Ex.633How many real roots does x⁵ − 5x + a have?

#polynomials #derivative #number of roots #parameter

Ex.1109Composition of polynomials

#composition #polynomials #degree

Ex.1110The discriminant of a quadratic

#discriminant #quadratic #roots

Ex.1111Divisibility of polynomials

#divisibility #roots #factorisation

Ex.1112Factorization of a cubic

#cubic #factorization #roots

Ex.1113Multiplicity of a root

#multiplicity #repeated root #derivative

Ex.1114The rational root test

#rational root theorem #factoring #roots

Ex.1115Factorization over $\mathbb{R}$

#factorization #real roots #complex roots

Ex.1116Sum of roots of unity

#roots of unity #sum #cyclotomic

Ex.1117Vieta's formulas for a cubic

#Vieta #cubic #symmetric functions

Ex.1231Divisibility of values of an integer polynomial

#integer polynomials #a−b divides P(a)−P(b) #integer roots

Ex.1292No polynomial takes only prime values

#integer polynomials #primes #divisibility #congruences

Ex.1293A monic polynomial with root √2 + √3 + √5

#algebraic numbers #minimal polynomial #rational root theorem #irrationality

Ex.1296Finite differences and alternating binomial sums

#finite differences #polynomials #binomial coefficients #degree

Ex.1298P(x²) = P(x)²

#polynomial equation #coefficients #degree #comparing terms

Ex.1302The derivative keeps all roots real

#polynomials #Rolle's theorem #multiplicity #real roots

Ex.1303P(x² + 1) = P(x)² + 1

#polynomial equation #recursive sequences #identity theorem

Ex.1818Bézout's identity for polynomials

#Bézout #polynomials #linear combination

Ex.1819Complex factorization via roots of unity

#roots of unity #factorization #complex

Ex.1820Decomposition into irreducible factors

#factorization #irreducible factors #real polynomials

Ex.1821A polynomial identity

#polynomial identity #factorization #symmetric

Ex.1822GCD of two polynomials

#GCD #Euclidean algorithm #polynomials

Ex.1823Lagrange interpolation

#interpolation #Lagrange #polynomial

Ex.1824Irreducibility of a polynomial

#irreducibility #polynomials #roots

Ex.1825Roots and coefficients

#roots #coefficients #bounds

Ex.1826A polynomial with integer coefficients

#integer coefficients #roots #Gauss lemma

Ex.1827A polynomial with parameter

#parameter #roots #Vieta

Ex.1828Rolle's theorem for polynomials

#Rolle #roots #derivatives

Ex.1829Roots of a cubic and discriminant

#cubic #discriminant #roots

Ex.1830Symmetric functions of roots

#symmetric functions #Vieta #Newton identities

Ex.1933Irreducibility with Eisenstein's criterion

#Eisenstein #irreducibility #cyclotomic polynomial #shift trick

Ex.1934A polynomial that equals k/(k+1) at 0, 1, …, n

#polynomial interpolation #degree argument #Lagrange

Ex.1936Polynomials with P(x²) = P(x)²

#polynomial equation #degree argument #coefficients comparison

Ex.1969P(x)P(x − 1) = P(x²)

#polynomial equation #roots #unit circle #functional equation

Ex.1971A polynomial that equals 1 / C(n+1, k) at k = 0, …, n

#polynomials #interpolation #finite differences #binomial coefficients

Ex.2188Cyclotomic polynomials

#cyclotomic #roots of unity #irreducibility

Ex.2189Common roots of two polynomials

#resultant #common root #gcd

Ex.2190Newton's identities for three variables

#Newton identities #symmetric functions #power sums

Ex.2225Chebyshev polynomials and rational values of cosine

#Chebyshev polynomials #algebraic integers #rational root theorem #Niven