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Polynomial division and the remainder theorem

Long division, synthetic division, and why f(a) is the remainder on dividing by x − a.

Dividing polynomials works like long division of integers: divide the leading terms, multiply back, subtract, repeat. The remainder theorem says dividing f(x) by x − a leaves remainder f(a), so a is a root exactly when the division is clean. Picture it: the quotient's graph is the original with the root's factor removed. Think it: this is the Euclidean algorithm again, over polynomials instead of integers.

Намунаи коркардашуда · simplify (x^3 - 1)/(x - 1)

Algebra · Simplifying

Simplify (x^3 - 1)/(x - 1)

\frac{x^{3} - 1}{x - 1}

Қадами ба қадам

  1. \frac{x^{3} - 1}{x - 1}

    Start from the expression.

  2. x^{2} + x + 1

    Factor numerator and denominator and cancel common factors.

Ҷавоби ҷавобро нишон диҳед
x^{2} + x + 1

Ҳоло шумо Масъаларо интихоб кунед, ё ки онро навишта ё кашед. Ҳар қадам, тасвир, ҷавоб то он даме, ки шумо пурсед, пинҳон мешавад.

Кор кардан

Дар ҳисоби ройгон ба ҳар як дарс қайдҳо илова карда мешаванд, қайди он, ки шумо чӣ кор кардаед, масъалаҳои ҳалшуда дар як ҷо ва муаллиме, ки шумо метавонед дар бораи ин саҳифа пурсед. Математика барои ҳама кушода аст, хоҳ ворид шуда бошад ё на.

Бақайдгирӣ Ворид шудан

Символы, используемые здесь

Барои муайянкунии пурраи маъно, тасвир ва маънои ҳар як ҳарф дар он, ба ҳар як аломат пахш кунед.

Чӣ тавр: Polynomial division and the remainder theorem

  1. Start from the expression.
  2. Factor numerator and denominator and cancel common factors.

Саволҳои маъмул

What is a function, really?

A rule that assigns exactly one output to each input. The vertical-line test on a graph is the same idea: no input may have two outputs.

Why do we need complex numbers?

Because x² + 1 = 0 has no real solution, and allowing one new number i with i² = −1 makes every polynomial equation solvable. They then turn out to describe rotation, waves and alternating current more naturally than real numbers do.

& Иловаи забон Precalculus

Complex numbersPolynomial functionsRational functionsSequences and seriesThe binomial theoremConic sectionsVectorsExponential and logarithmic functionsParametric equations and polar coordinates