| Deliver and maintain services, like tracking outages and protecting against spam, fraud, and abuse• See for a theoretical explanation of why this is so, and see for ways to approximate roots of polynomials numerically | But since we have been using a method which finds only rational roots, it is very likely that R x is not equal to P x ; it is very likely that P x has some irrational or complex roots not in R |
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| Note: by a "complete factorization" of P x over Q, we mean a factorization as a product of polynomials with rational coefficients, such that each factor is irreducible over Q, where "irreducible over Q" means that the factor cannot be written as the product of two non-constant polynomials with rational coefficients and smaller degree | a 1 a 0 r b n-1r |
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9| a 1 a 0 r ---- ---------- 2 | |
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| Personalized content and ads include things like more relevant results and recommendations, a customized YouTube homepage, and ads that are tailored to your interests | But if we were asked to "factor into irreducibles over Q", we might think we are done; but it is important to realize that this might not necessarily be the case |
Google uses and data to:• Improve the quality of our services and develop new ones• Personalized content and ads can be based on those things and your activity like Google searches and videos you watch on YouTube.
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