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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.
कार्यरत उदाहरण · simplify (x^3 - 1)/(x - 1)
चरणानुक्रमे
- \frac{x^{3} - 1}{x - 1}
Start from the expression.
- x^{2} + x + 1
Factor numerator and denominator and cancel common factors.
उत्तर उघडा
आता तुम्ही प्रश्न निवडा, किंवा स्वतःचा प्रश्न टाइप करा किंवा रेखाटून दाखवा. प्रत्येक टप्प्यात, चित्रे आहेत, तुम्ही विचारण्याआधी उत्तर लपवलेले आहे.
मोफत खाते प्रत्येक शिकवणी वर टिपण्णी जोडते, तुम्ही पूर्ण केलेल्या गोष्टीचा नोंदवही, तुमच्या सोडविलेल्या समस्या एका ठिकाणी, आणि या पानाच्या विषयी तुम्ही विचारू शकता अशा शिक्षकाला. गणित स्वतःच सर्वांसाठी खुले आहे, दाखलन झालेले असो किंवा नसो.
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कसे कराल: Polynomial division and the remainder theorem
- Start from the expression.
- Factor numerator and denominator and cancel common factors.
लोक विचारतात ते प्रश्न
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.
अधिक माहिती Precalculus
Complex numbersPolynomial functionsRational functionsSequences and seriesThe binomial theoremConic sectionsVectorsExponential and logarithmic functionsParametric equations and polar coordinates