19. We say that a e Fis a multiple root of f(3) = FE (- a* isa factor of x) for some kz 2. ) Prove that a eR is a multiple rootof ftx) E Rl if and only if e is a root of both f(x) andƒ(x), where f"(x) is the derivative of f(x). (6) T (*) E RE and if E) is relatively prime to S (8), prove that /(x) has no multiple root in R.
Family of Curves
A family of curves is a group of curves that are each described by a parametrization in which one or more variables are parameters. In general, the parameters have more complexity on the assembly of the curve than an ordinary linear transformation. These families appear commonly in the solution of differential equations. When a constant of integration is added, it is normally modified algebraically until it no longer replicates a plain linear transformation. The order of a differential equation depends on how many uncertain variables appear in the corresponding curve. The order of the differential equation acquired is two if two unknown variables exist in an equation belonging to this family.
XZ Plane
In order to understand XZ plane, it's helpful to understand two-dimensional and three-dimensional spaces. To plot a point on a plane, two numbers are needed, and these two numbers in the plane can be represented as an ordered pair (a,b) where a and b are real numbers and a is the horizontal coordinate and b is the vertical coordinate. This type of plane is called two-dimensional and it contains two perpendicular axes, the horizontal axis, and the vertical axis.
Euclidean Geometry
Geometry is the branch of mathematics that deals with flat surfaces like lines, angles, points, two-dimensional figures, etc. In Euclidean geometry, one studies the geometrical shapes that rely on different theorems and axioms. This (pure mathematics) geometry was introduced by the Greek mathematician Euclid, and that is why it is called Euclidean geometry. Euclid explained this in his book named 'elements'. Euclid's method in Euclidean geometry involves handling a small group of innately captivate axioms and incorporating many of these other propositions. The elements written by Euclid are the fundamentals for the study of geometry from a modern mathematical perspective. Elements comprise Euclidean theories, postulates, axioms, construction, and mathematical proofs of propositions.
Lines and Angles
In a two-dimensional plane, a line is simply a figure that joins two points. Usually, lines are used for presenting objects that are straight in shape and have minimal depth or width.
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(b) Supposc r, s, t e Farc roots of ax + bx + ex + de F[x] (with a + Op).
Show that r + s + t = -a-b and rs + st + rt = a-'cand rst = -a'd.
15. Prove that +1 is reducible in Z[x] if and only if there exist integers a and b
such that p = a + b and ab = 1 (mod p).
16. Let f(x), g(x) E F[x] have degree sn and let co, C1, . . . , C, be distinct elements
of F. If f(c) = g(c) for i = 0, 1, ..., n, prove that f(x) = g(x) in F[x].
17. Find a polynomial of degree 2 in Zdx] that has four roots in Z. Does this
contradict Corollary 4.17?
18. Let ç:C → C be an isomorphism of rings such that o(a) = a for each
a E Q. Suppose r e C isa root of f(x) E Q[x]. Prove that o(r) is also a
root of f(x).
19. We say that a E Fis a multiple root of f(x) E F[x] if (x - a)* is a factor of
f(x) for some k > 2.
(a) Prove that a ER is a multiple root of f(x) e R[x] if and only if « is a
root of both f(x) and f'(x), where f"(x) is the derivative of f(x).
(b) If f(x) E R[x] and if f(x) is relatively prime to f"(x), prove that f(x) has
no multiple root in R.
20. Let R be an integral domain. Then the Division Algorithm holds in R[x]
whenever the divisor is monic, by Exercise 14 in Section 4.1. Use this fact to
show that the Remainder and Factor Theorems hold in R[x].
21. If R is an integral domain and f(x) is a nonzero polynomial of degree n in
R[x], prove that f(x) has at most n roots in R. [Hint: Exercise 20.]
22. Show that Corollary 4.20 holds if Fis an infinite integral domain. [Hint: See
Exercise 21.]
23. Let f(x), g(x), h(x) E F[x] and re F.
(a) If f(x) = g(x) + h(x) in F[x], show that f(r) = g(r) + h(r) in F.
(b) If f(x) = g(x)h(x) in F[x], show that f(r) = g(r)h(r) in F.
Where were these facts used in this section?
24. Let a be a fixed element of Fand define a map 4.¿F[x] →Fby Pa[f(x)] = f(a).
Prove that o, is a surjective homomorphism of rings The map q, is called an
evaluation homomorphism; there is one for each a e F.
25. Let Q[T] be the set of all real numbers of the form
ro + riT + r2 + ·.·+ a,", with n20 and r, E Q.
(a) Show that Q[T]is a subring of R.
(b) Show that the function 0:Q[x] –→Q[7] defined by ®( f(x)) = f(T) is an
isomorphism. You may assume the following nontrivial fact: 7 is not
the root of any nonzero polynomial with rational coefficients. Therefore,
Theorem 4.1 is true with R = Q and T in place of x. However, see
Exercise 26.
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