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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.
ຕົວຢ່າງທີ່ໄດ້ເຮັດ · simplify (x^3 - 1)/(x - 1)
ຂັ້ນຕອນຕໍ່ຂັ້ນຕອນ
- \frac{x^{3} - 1}{x - 1}
Start from the expression.
- x^{2} + x + 1
Factor numerator and denominator and cancel common factors.
ເປີດເຜີຍຄຳຕອບ
ປັດຈຸບັນທ່ານ ເລືອກບັນຫາ ຫຼື ພິມ ຫຼື ແຕ້ມເອງ. ທຸກບາດກ້າວ, ຮູບ, ຄໍາຕອບຖືກລົບຈົນກວ່າເຈົ້າຈະຖາມມັນ.
ຕົວສະແດງທີ່ໃຊ້ຢູ່ທີ່ນີ້
ກົດຕົວອັກສອນໃດໜຶ່ງເພື່ອເບິ່ງຄວາມໝາຍເຕັມ, ຮູບ ແລະ ຕົວອັກສອນແຕ່ລະຕົວໃນມັນໝາຍຄວາມວ່າແນວໃດ.
ວິທີການ: Polynomial division and the remainder theorem
- Start from the expression.
- Factor numerator and denominator and cancel common factors.
ຄໍາຖາມທີ່ຄົນຖາມ
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.
ເພີ່ມເຕີມໃນ Precalculus
Complex numbersPolynomial functionsRational functionsSequences and seriesThe binomial theoremConic sectionsVectorsExponential and logarithmic functionsParametric equations and polar coordinates