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25 Jan 2026

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5 pages

Tot sobre Polinomis: Definicions i Exercicis

German Camp Vallès

@germancampvalls_twms

Els polinomis són expressions algebraiques que combinen nombres i lletres... Show more

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2.4. POLINOMIS (p.30)

Conceptes previs:
Polinomi = P(x) és una expressió algebraica formada per la combinació de nombres i lletres units pe

Conceptes Bàsics dels Polinomis

Els polinomis són com receptes matemàtiques que combinen nombres i lletres. Un monomi és la peça més petita (com 3x²), mentre que un polinomi és la combinació de diversos monomis units per sumes o restes.

La variable és la lletra misteriosa (normalment x) i el coeficient és el nombre que la multiplica. Per exemple, en 5x³, el 5 és el coeficient i x³ és la part literal. Quan dos monomis tenen exactament la mateixa part literal, els anomenem monomis semblants.

El grau d'un monomi és la suma de tots els exponents de les seves variables. El grau d'un polinomi sencer és el grau del seu terme més gran. El terme independent és aquell que no té variables, només un nombre.

Consell clau: Per trobar el valor numèric d'un polinomi, només has de substituir les lletres pels nombres donats i calcular!

Operacions amb Monomis i Polinomis

Sumar i restar monomis és súper fàcil si són semblants: només sumes els coeficients i mantens la mateixa part literal. Per multiplicar monomis, multipliques els coeficients i sumes els exponents de les mateixes variables.

Per a les potències, elevas el coeficient a l'exponent i multipliques tots els exponents de les variables per aquest número. La propietat distributiva és el teu millor amic: a $b + c$ = ab + ac.

Les identitats notables són fórmules màgiques que t'estalvien temps:

$a + b$ ² = a² + 2ab + b²
$a - b$ ² = a² - 2ab + b²
$a + b$ $a - b$ = a² - b²

Truc d'examen: Memoritza aquestes identitats notables - apareixen constantment als exàmens i t'estalvien molt temps de càlcul!

Divisió de Polinomis

Dividir polinomis és com fer una divisió llarga, però amb lletres. Quan divideixes P(x) entre Q(x), obtens un quocient C(x) i un residu R(x), que compleixen: P(x) = Q(x) · C(x) + R(x).

El grau del dividend ha de ser major o igual que el del divisor. El grau del quocient sempre és la diferència entre els graus del dividend i el divisor. Si el residu és zero, la divisió és exacta.

Els exemples mostren com organitzar els càlculs pas a pas. Primer divideixes els termes de major grau, després multipliques cap enrere i restes fins que no puguis continuar.

Verificació important: Sempre comprova la teva resposta substituint en la fórmula P(x) = Q(x) · C(x) + R(x)!

Regla de Ruffini i Teoremes Fonamentals

La Regla de Ruffini és un mètode ràpid per dividir polinomis quan el divisor té la forma $x - a$ . És molt més eficient que la divisió llarga tradicional i t'estalvia molta feina als exàmens.

El teorema del residu és genial: et diu que per trobar el residu de dividir P(x) entre $x - a$ , només has de calcular P(a). Les arrels d'un polinomi són els valors que fan que el polinomi sigui igual a zero.

Una propietat súper útil: les possibles arrels enteres d'un polinomi sempre són divisors del terme independent. El nombre total d'arrels mai supera el grau del polinomi.

Estratègia d'examen: Per trobar arrels ràpidament, prova primer amb els divisors del terme independent: ±1, ±2, ±3...

Factorització de Polinomis

Factoritzar un polinomi significa descomposar-lo en un producte de factors més simples. Si un polinomi té n arrels reals, es pot escriure com P(x) = aₙ $x - r₁$ $x - r₂$ ... $x - rₙ$ .

Els mètodes principals per factoritzar són: treure factor comú, aplicar identitats notables, resoldre equacions de segon grau, i trobar arrels amb Ruffini per a polinomis de grau superior.

La factorització és l'objectiu final de molts problemes de polinomis. Una vegada factoritzat, pots resoldre equacions, simplificar fraccions algebraiques i entendre millor el comportament del polinomi.

Consell pràctic: Sempre comença intentant treure factor comú abans de buscar arrels o aplicar altres mètodes - és el més ràpid!

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Matemáticas

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296

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25 Jan 2026

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5 pages

Tot sobre Polinomis: Definicions i Exercicis

German Camp Vallès

@germancampvalls_twms

Els polinomis són expressions algebraiques que combinen nombres i lletres mitjançant operacions matemàtiques bàsiques. Dominar-los és essencial per a l'àlgebra avançada i molts exàmens de matemàtiques, ja que apareixen constantment en equacions i càlculs complexos.

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Conceptes Bàsics dels Polinomis

Consell clau: Per trobar el valor numèric d'un polinomi, només has de substituir les lletres pels nombres donats i calcular!

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Operacions amb Monomis i Polinomis

Les identitats notables són fórmules màgiques que t'estalvien temps:

$a + b$ ² = a² + 2ab + b²
$a - b$ ² = a² - 2ab + b²
$a + b$ $a - b$ = a² - b²

Truc d'examen: Memoritza aquestes identitats notables - apareixen constantment als exàmens i t'estalvien molt temps de càlcul!

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Divisió de Polinomis

Dividir polinomis és com fer una divisió llarga, però amb lletres. Quan divideixes P(x) entre Q(x), obtens un quocient C(x) i un residu R(x), que compleixen: P(x) = Q(x) · C(x) + R(x).

Els exemples mostren com organitzar els càlculs pas a pas. Primer divideixes els termes de major grau, després multipliques cap enrere i restes fins que no puguis continuar.

Verificació important: Sempre comprova la teva resposta substituint en la fórmula P(x) = Q(x) · C(x) + R(x)!

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Regla de Ruffini i Teoremes Fonamentals

Una propietat súper útil: les possibles arrels enteres d'un polinomi sempre són divisors del terme independent. El nombre total d'arrels mai supera el grau del polinomi.

Estratègia d'examen: Per trobar arrels ràpidament, prova primer amb els divisors del terme independent: ±1, ±2, ±3...

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Improve your grades

Join milions of students

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Factorització de Polinomis

Factoritzar un polinomi significa descomposar-lo en un producte de factors més simples. Si un polinomi té n arrels reals, es pot escriure com P(x) = aₙ $x - r₁$ $x - r₂$ ... $x - rₙ$ .

Els mètodes principals per factoritzar són: treure factor comú, aplicar identitats notables, resoldre equacions de segon grau, i trobar arrels amb Ruffini per a polinomis de grau superior.

Consell pràctic: Sempre comença intentant treure factor comú abans de buscar arrels o aplicar altres mètodes - és el més ràpid!