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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.

လုပ်ဆောင်ခဲ့သောဥပမာ: simplify (x^3 - 1)/(x - 1)

Algebra · Simplifying

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

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

ခြေလှမ်းတစ်လှမ်း

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

    Start from the expression.

  2. x^{2} + x + 1

    Factor numerator and denominator and cancel common factors.

အဖြေကို ဖော်ပြပါ
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.

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ပိုပြီး Precalculus

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