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Chapter 3: Rational and Real Numbers
Chapter 3: Rational and Real Numbers

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On the Relation between Polynomial Identity Testing and Finding

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... A quadratic factor with no real zeros is said to be prime or irreducible over the reals. Be sure you see that this is not the same as being irreducible over the rationals. For example, the quadratic x2 + 1 = (x – i)(x + i) is irreducible over the reals (and therefore over the rationals). ...
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... with the other two denominators? Needs an ‘x’ 4. What does the 3rd denominator need to be common with the other two denominators? ...
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... 7.NS.1.b Understand p + q as the number located a distance |q| from p, in the positive or negative direction depending on whether q is positive or negative. Show that a number and its opposite have a sum of 0 (are additive inverses). Interpret sums of rational numbers by describing real-world contex ...
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LU decomposition - National Cheng Kung University

... One of the other uses of norms is to establish the convergence ...
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... Exercise 3 shows why the mnemonic FOIL is not very helpful—and in this case does not make sense. By now, students should have had enough practice multiplying to no longer require such mnemonics to help them. They understand that the multiplications they are doing are really repeated use of the distr ...
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... That is, an equation can be multiplied or divided by the same nonzero real number without changing the solution of the equation. Example: Solve 2x + 7 = 19. Original equation 2x + 7 = 19 The solution is preserved when 7 is 2x + 7  7 = 19  7 subtracted from both sides. 2x = 12 Simplify both sides. ...
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Unification for infinite sets of equations between finite terms
Unification for infinite sets of equations between finite terms

... The unification algorithms work only for finite sets of equations. For example, the infinite collection {xi = xi+1 | i = 0, 1, 2, ...} has most general unifiers σn for n = 0, 1, 2, ... with (xi )σn ≡ xn for i = 0, 1, 2, .... The unification algorithms would not terminate in the process of computing ...
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System of polynomial equations

A system of polynomial equations is a set of simultaneous equations f1 = 0, ..., fh = 0 where the fi are polynomials in several variables, say x1, ..., xn, over some field k.Usually, the field k is either the field of rational numbers or a finite field, although most of the theory applies to any field.A solution is a set of the values for the xi which make all of the equations true and which belong to some algebraically closed field extension K of k. When k is the field of rational numbers, K is the field of complex numbers.
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