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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.

Mfano wenye matokeo: simplify (x^3 - 1)/(x - 1)

Algebra · Simplifying

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

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

Hatua kwa hatua

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

    Start from the expression.

  2. x^{2} + x + 1

    Factor numerator and denominator and cancel common factors.

Lafunua jibu
x^{2} + x + 1

Symbols used here

|x|
absolute value / modulus
Distance from zero: |−3| = 3. For a complex number, distance from the origin.
\binom{n}{k}
binomial coefficient, "n choose k"
Number of k-element subsets of n things: n!/(k!(n−k)!).
\pi
pi
Ratio of a circle's circumference to its diameter, 3.14159…
e
Euler's number
2.71828…, the base whose exponential is its own derivative.
i
imaginary unit
i² = −1.
\theta
theta
The usual name for an angle.
\log_b x,\ \ln x
logarithm, natural log
The exponent b must be raised to for x; ln uses base e.
\mathbf{v},\ \vec{v}
vector
A quantity with magnitude and direction; a column of numbers.

How to: Polynomial division and the remainder theorem

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

Questions people ask

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.

Jaribu kufanya mambo yako mwenyewe

Mengi zaidi katika Precalculus

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