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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.
Работен пример · simplify (x^3 - 1)/(x - 1)
Стъпка по стъпка
- \frac{x^{3} - 1}{x - 1}
Start from the expression.
- x^{2} + x + 1
Factor numerator and denominator and cancel common factors.
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Как да се: Polynomial division and the remainder theorem
- Start from the expression.
- Factor numerator and denominator and cancel common factors.
Въпроси, които хората задават
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.
Още в Precalculus
Complex numbersPolynomial functionsRational functionsSequences and seriesThe binomial theoremConic sectionsVectorsExponential and logarithmic functionsParametric equations and polar coordinates