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The polynomial P = x 4 + 1 is irreducible over Q but not over any finite field. On any field extension of F 2, P = (x + 1) 4. On every other finite field, at least one of −1, 2 and −2 is a square, because the product of two non-squares is a square and so we have; If =, then = (+) ().
For example, 3 × 5 is an integer factorization of 15, and (x – 2)(x + 2) is a polynomial factorization of x 2 – 4. Factorization is not usually considered meaningful within number systems possessing division , such as the real or complex numbers , since any x {\displaystyle x} can be trivially written as ( x y ) × ( 1 / y ) {\displaystyle ...
If one of these values is 0, we have a linear factor. If the values are nonzero, we can list the possible factorizations for each. Now, 2 can only factor as 1×2, 2×1, (−1)×(−2), or (−2)×(−1). Therefore, if a second degree integer polynomial factor exists, it must take one of the values p(0) = 1, 2, −1, or −2. and likewise for p(-1
Squares are always congruent to 0, 1, 4, 5, 9, 16 modulo 20. The values repeat with each increase of a by 10. In this example, N is 17 mod 20, so subtracting 17 mod 20 (or adding 3), produces 3, 4, 7, 8, 12, and 19 modulo 20 for these values. It is apparent that only the 4 from this list can be a square.
The polynomial x 2 + 1 = 0 has roots ± i. Any real square matrix of odd degree has at least one real eigenvalue. For example, if the matrix is orthogonal, then 1 or −1 is an eigenvalue. The polynomial + has roots , +,, and thus can be factored as
Concrete mixer. A concrete mixer (also cement mixer) is a device that homogeneously combines cement, ... RCC and SCC, typically in batches of 2–6 m 3 (2.6–7.8 cu yd).
The volumetric mixer varies in capacity size up to 12 m 3 and has a production rate of around 60m 3 an hour depending on the mix design. Many volumetric concrete mixer manufacturers have innovated the mixer in capacity and design, as well as added features including color, multiple admixes, fiber systems, and the ability to do gunite or shotcrete.
Cement hydration and strength development mainly depend on two silicate phases: tricalcium silicate (C 3 S) , and dicalcium silicate (C 2 S) . [1] Upon hydration, the main reaction products are calcium silicate hydrates (C-S-H) and calcium hydroxide Ca(OH) 2, written as CH in the cement chemist notation. C-S-H is the phase playing the role of ...