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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.

tusaale · simplify (x^3 - 1)/(x - 1)

Algebra · Simplifying

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

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

tallaabo tallaabo

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

    Start from the expression.

  2. x^{2} + x + 1

    Factor numerator and denominator and cancel common factors.

Jawaabta muuji
x^{2} + x + 1

Haddaad Dooro dhibaato, ama qoraal ama sawirkaaga. Talaab kasta, sawir, jawaabta qarsoon ilaa aad weydiiso.

Ku hay shaqadaada

A xisaab bilaash ah oo ku daray qoraalada waxbarasho kasta, diiwaanka waxa aad dhamaatay, dhibaatooyinka aad xal u tahay meel kaliya, iyo macallinka aad weydiisan kartaa oo ku saabsan bogga. Xisaabta oo keliya waa u furan yahay qof kasta, la diiwaan geliyey ama aan.

Ku soo biir Ku soo gal

Calaamadaha la isticmaalay halkan

Taab calaamad kasta oo loogu talagalay qeexidda buuxda, sawir, iyo waxa xarfaha oo dhan ka mid ah ka dhigan tahay.

Sida loo: Polynomial division and the remainder theorem

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

Su'aalaha dadku waydiiyaan

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.

In ka badan Precalculus

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