Intermediate Algebra
Intermediate Algebra
10th Edition
ISBN: 9781285195728
Author: Jerome E. Kaufmann, Karen L. Schwitters
Publisher: Cengage Learning
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Chapter 8.2, Problem 43PS
To determine

To plot:

The graph of circle x2+y2+4x+4y8=0.

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1.2.19. Let and s be natural numbers. Let G be the simple graph with vertex set Vo... V„−1 such that v; ↔ v; if and only if |ji| Є (r,s). Prove that S has exactly k components, where k is the greatest common divisor of {n, r,s}.
Question 3 over a field K. In this question, MË(K) denotes the set of n × n matrices (a) Suppose that A Є Mn(K) is an invertible matrix. Is it always true that A is equivalent to A-¹? Justify your answer. (b) Let B be given by 8 B = 0 7 7 0 -7 7 Working over the field F2 with 2 elements, compute the rank of B as an element of M2(F2). (c) Let 1 C -1 1 [4] [6] and consider C as an element of M3(Q). Determine the minimal polynomial mc(x) and hence, or otherwise, show that C can not be diagonalised. [7] (d) Show that C in (c) considered as an element of M3(R) can be diagonalised. Write down all the eigenvalues. Show your working. [8]
R denotes the field of real numbers, Q denotes the field of rationals, and Fp denotes the field of p elements given by integers modulo p. You may refer to general results from lectures. Question 1 For each non-negative integer m, let R[x]m denote the vector space consisting of the polynomials in x with coefficients in R and of degree ≤ m. x²+2, V3 = 5. Prove that (V1, V2, V3) is a linearly independent (a) Let vi = x, V2 = list in R[x] 3. (b) Let V1, V2, V3 be as defined in (a). Find a vector v € R[×]3 such that (V1, V2, V3, V4) is a basis of R[x] 3. [8] [6] (c) Prove that the map ƒ from R[x] 2 to R[x]3 given by f(p(x)) = xp(x) — xp(0) is a linear map. [6] (d) Write down the matrix for the map ƒ defined in (c) with respect to the basis (2,2x + 1, x²) of R[x] 2 and the basis (1, x, x², x³) of R[x] 3. [5]

Chapter 8 Solutions

Intermediate Algebra

Ch. 8.1 - Prob. 1PSCh. 8.1 - Prob. 2PSCh. 8.1 - Prob. 3PSCh. 8.1 - Prob. 4PSCh. 8.1 - Prob. 5PSCh. 8.1 - Prob. 6PSCh. 8.1 - Prob. 7PSCh. 8.1 - Prob. 8PSCh. 8.1 - Prob. 9PSCh. 8.1 - Prob. 10PSCh. 8.1 - Prob. 11PSCh. 8.1 - Prob. 12PSCh. 8.1 - Prob. 13PSCh. 8.1 - Prob. 14PSCh. 8.1 - Prob. 15PSCh. 8.1 - Prob. 16PSCh. 8.1 - Prob. 17PSCh. 8.1 - Prob. 18PSCh. 8.1 - Prob. 19PSCh. 8.1 - Prob. 20PSCh. 8.1 - Prob. 21PSCh. 8.1 - Prob. 22PSCh. 8.1 - Prob. 23PSCh. 8.1 - Prob. 24PSCh. 8.1 - Prob. 25PSCh. 8.1 - Prob. 26PSCh. 8.1 - Prob. 27PSCh. 8.1 - Prob. 28PSCh. 8.1 - Prob. 29PSCh. 8.1 - Prob. 30PSCh. 8.1 - Prob. 31PSCh. 8.1 - Prob. 32PSCh. 8.1 - Prob. 33PSCh. 8.1 - Prob. 34PSCh. 8.1 - Prob. 35PSCh. 8.1 - Prob. 36PSCh. 8.1 - Prob. 37PSCh. 8.1 - Prob. 38PSCh. 8.1 - Prob. 39PSCh. 8.2 - Prob. 1CQCh. 8.2 - Prob. 2CQCh. 8.2 - Prob. 3CQCh. 8.2 - Prob. 4CQCh. 8.2 - Prob. 5CQCh. 8.2 - Prob. 6CQCh. 8.2 - Prob. 7CQCh. 8.2 - Prob. 8CQCh. 8.2 - Prob. 9CQCh. 8.2 - Prob. 10CQCh. 8.2 - Prob. 1PSCh. 8.2 - Prob. 2PSCh. 8.2 - Prob. 3PSCh. 8.2 - Prob. 4PSCh. 8.2 - Prob. 5PSCh. 8.2 - Prob. 6PSCh. 8.2 - Prob. 7PSCh. 8.2 - Prob. 8PSCh. 8.2 - Prob. 9PSCh. 8.2 - Prob. 10PSCh. 8.2 - Prob. 11PSCh. 8.2 - Prob. 12PSCh. 8.2 - Prob. 13PSCh. 8.2 - Prob. 14PSCh. 8.2 - Prob. 15PSCh. 8.2 - Prob. 16PSCh. 8.2 - Prob. 17PSCh. 8.2 - Prob. 18PSCh. 8.2 - Prob. 19PSCh. 8.2 - Prob. 20PSCh. 8.2 - Prob. 21PSCh. 8.2 - Prob. 22PSCh. 8.2 - Prob. 23PSCh. 8.2 - Prob. 24PSCh. 8.2 - Prob. 25PSCh. 8.2 - Prob. 26PSCh. 8.2 - Prob. 27PSCh. 8.2 - Prob. 28PSCh. 8.2 - Prob. 29PSCh. 8.2 - Prob. 30PSCh. 8.2 - Prob. 31PSCh. 8.2 - Prob. 32PSCh. 8.2 - Prob. 33PSCh. 8.2 - Prob. 34PSCh. 8.2 - Prob. 35PSCh. 8.2 - Prob. 36PSCh. 8.2 - Prob. 37PSCh. 8.2 - Prob. 38PSCh. 8.2 - Prob. 39PSCh. 8.2 - Prob. 40PSCh. 8.2 - Prob. 41PSCh. 8.2 - Prob. 42PSCh. 8.2 - Prob. 43PSCh. 8.2 - Prob. 44PSCh. 8.2 - Prob. 45PSCh. 8.2 - Prob. 46PSCh. 8.2 - Prob. 47PSCh. 8.2 - Prob. 48PSCh. 8.2 - Prob. 49PSCh. 8.2 - Prob. 50PSCh. 8.2 - Prob. 51PSCh. 8.2 - Prob. 52PSCh. 8.2 - Prob. 53PSCh. 8.2 - Prob. 54PSCh. 8.2 - Prob. 55PSCh. 8.2 - Prob. 56PSCh. 8.2 - Prob. 57PSCh. 8.2 - Prob. 58PSCh. 8.2 - Prob. 59PSCh. 8.2 - Prob. 60PSCh. 8.2 - Prob. 61PSCh. 8.2 - Prob. 62PSCh. 8.2 - Prob. 63.1PSCh. 8.2 - By expanding (xh)2+(yk)2=r2, we obtain...Ch. 8.2 - Prob. 63.3PSCh. 8.2 - Prob. 63.4PSCh. 8.2 - Prob. 63.5PSCh. 8.2 - Prob. 63.6PSCh. 8.2 - Prob. 64PSCh. 8.2 - Prob. 65PSCh. 8.2 - Prob. 66.1PSCh. 8.2 - Prob. 66.2PSCh. 8.2 - Prob. 66.3PSCh. 8.2 - Prob. 66.4PSCh. 8.2 - Prob. 66.5PSCh. 8.2 - Prob. 66.6PSCh. 8.3 - Prob. 1CQCh. 8.3 - Prob. 2CQCh. 8.3 - Prob. 3CQCh. 8.3 - Prob. 4CQCh. 8.3 - Prob. 5CQCh. 8.3 - Prob. 6CQCh. 8.3 - Prob. 7CQCh. 8.3 - Prob. 8CQCh. 8.3 - Prob. 9CQCh. 8.3 - Prob. 10CQCh. 8.3 - Prob. 1PSCh. 8.3 - Prob. 2PSCh. 8.3 - Prob. 3PSCh. 8.3 - Prob. 4PSCh. 8.3 - Prob. 5PSCh. 8.3 - Prob. 6PSCh. 8.3 - Prob. 7PSCh. 8.3 - Prob. 8PSCh. 8.3 - Prob. 9PSCh. 8.3 - Prob. 10PSCh. 8.3 - Prob. 11PSCh. 8.3 - Prob. 12PSCh. 8.3 - Prob. 13PSCh. 8.3 - Prob. 14PSCh. 8.3 - Prob. 15PSCh. 8.3 - Prob. 16PSCh. 8.3 - Prob. 17PSCh. 8.3 - Prob. 18PSCh. 8.3 - Prob. 19PSCh. 8.3 - Prob. 20PSCh. 8.3 - Prob. 21PSCh. 8.3 - Prob. 22PSCh. 8.3 - Prob. 23PSCh. 8.3 - Prob. 24PSCh. 8.3 - Prob. 25PSCh. 8.3 - Prob. 26PSCh. 8.3 - Prob. 27PSCh. 8.3 - Prob. 28PSCh. 8.3 - Prob. 29PSCh. 8.3 - Prob. 30PSCh. 8.4 - Prob. 1CQCh. 8.4 - Prob. 2CQCh. 8.4 - Prob. 3CQCh. 8.4 - Prob. 4CQCh. 8.4 - Prob. 5CQCh. 8.4 - Prob. 6CQCh. 8.4 - Prob. 7CQCh. 8.4 - Prob. 8CQCh. 8.4 - Prob. 9CQCh. 8.4 - Prob. 10CQCh. 8.4 - Prob. 1PSCh. 8.4 - Prob. 2PSCh. 8.4 - Prob. 3PSCh. 8.4 - Prob. 4PSCh. 8.4 - Prob. 5PSCh. 8.4 - Prob. 6PSCh. 8.4 - Prob. 7PSCh. 8.4 - Prob. 8PSCh. 8.4 - Prob. 9PSCh. 8.4 - Prob. 10PSCh. 8.4 - Prob. 11PSCh. 8.4 - Prob. 12PSCh. 8.4 - Prob. 13PSCh. 8.4 - Prob. 14PSCh. 8.4 - Prob. 15PSCh. 8.4 - Prob. 16PSCh. 8.4 - Prob. 17PSCh. 8.4 - Prob. 18PSCh. 8.4 - Prob. 19PSCh. 8.4 - Prob. 20PSCh. 8.4 - Prob. 21PSCh. 8.4 - Prob. 22PSCh. 8.4 - Prob. 23PSCh. 8.4 - Prob. 24PSCh. 8.4 - Prob. 25PSCh. 8.4 - Prob. 26PSCh. 8.4 - Prob. 27PSCh. 8.4 - Prob. 28PSCh. 8.4 - Prob. 29PSCh. 8.4 - Prob. 30PSCh. 8.4 - Prob. 31PSCh. 8.4 - Prob. 32PSCh. 8.4 - Prob. 33PSCh. 8.4 - Prob. 34PSCh. 8.4 - Prob. 35PSCh. 8.4 - Prob. 36PSCh. 8.4 - Prob. 37PSCh. 8.4 - Prob. 38PSCh. 8.4 - Prob. 39PSCh. 8.4 - Prob. 40.1PSCh. 8.4 - Prob. 40.2PSCh. 8.4 - Prob. 40.3PSCh. 8.4 - Prob. 40.4PSCh. 8.4 - Prob. 40.5PSCh. 8.4 - Prob. 40.6PSCh. 8.4 - Prob. 41.1PSCh. 8.4 - Prob. 41.2PSCh. 8.4 - Prob. 41.3PSCh. 8.4 - Prob. 41.4PSCh. 8.4 - Prob. 41.5PSCh. 8.4 - Prob. 41.6PSCh. 8.4 - Prob. 41.7PSCh. 8.4 - Prob. 41.8PSCh. 8.4 - Prob. 41.9PSCh. 8.4 - Prob. 41.10PSCh. 8.4 - Prob. 42PSCh. 8.S - Prob. 1SCh. 8.S - Prob. 2SCh. 8.S - Prob. 3SCh. 8.S - Prob. 4SCh. 8.S - Prob. 5SCh. 8.S - Prob. 6SCh. 8.S - Prob. 7SCh. 8.S - Prob. 8SCh. 8.CR - Prob. 1CRCh. 8.CR - Prob. 2CRCh. 8.CR - Prob. 3CRCh. 8.CR - Prob. 4CRCh. 8.CR - Prob. 5CRCh. 8.CR - Prob. 6CRCh. 8.CR - Prob. 7CRCh. 8.CR - Prob. 8CRCh. 8.CR - Prob. 9CRCh. 8.CR - Prob. 10CRCh. 8.CR - Prob. 11CRCh. 8.CR - Prob. 12CRCh. 8.CR - Prob. 13CRCh. 8.CR - Prob. 14CRCh. 8.CR - Prob. 15CRCh. 8.CR - Prob. 16CRCh. 8.CR - Prob. 17CRCh. 8.CR - Prob. 18CRCh. 8.CR - Prob. 19CRCh. 8.CR - Prob. 20CRCh. 8.CR - Prob. 21CRCh. 8.CR - Prob. 22CRCh. 8.CR - Prob. 23CRCh. 8.CR - Prob. 24CRCh. 8.CR - Prob. 25CRCh. 8.CR - Prob. 26CRCh. 8.CR - Prob. 27CRCh. 8.CR - Prob. 28CRCh. 8.CR - Prob. 29CRCh. 8.CR - Prob. 30CRCh. 8.CR - Prob. 31CRCh. 8.CR - Prob. 32CRCh. 8.CR - Prob. 33CRCh. 8.CR - For Problems 3150, graph each equation....Ch. 8.CR - Prob. 35CRCh. 8.CR - Prob. 36CRCh. 8.CR - Prob. 37CRCh. 8.CR - Prob. 38CRCh. 8.CR - Prob. 39CRCh. 8.CR - Prob. 40CRCh. 8.CR - Prob. 41CRCh. 8.CR - Prob. 42CRCh. 8.CR - Prob. 43CRCh. 8.CR - Prob. 44CRCh. 8.CR - Prob. 45CRCh. 8.CR - Prob. 46CRCh. 8.CR - Prob. 47CRCh. 8.CR - Prob. 48CRCh. 8.CR - Prob. 49CRCh. 8.CR - Prob. 50CRCh. 8.CT - Prob. 1CTCh. 8.CT - Prob. 2CTCh. 8.CT - Prob. 3CTCh. 8.CT - Prob. 4CTCh. 8.CT - Prob. 5CTCh. 8.CT - Prob. 6CTCh. 8.CT - Prob. 7CTCh. 8.CT - Prob. 12CTCh. 8.CT - Prob. 13CTCh. 8.CT - Prob. 14CTCh. 8.CT - Prob. 15CTCh. 8.CT - Prob. 16CTCh. 8.CT - Prob. 17CTCh. 8.CT - Prob. 18CTCh. 8.CT - Prob. 19CTCh. 8.CT - Prob. 20CTCh. 8.CT - Prob. 21CTCh. 8.CT - Prob. 22CTCh. 8.CT - Prob. 23CTCh. 8.CT - Prob. 24CTCh. 8.CT - Prob. 25CTCh. 8.CM - Prob. 1CMCh. 8.CM - Prob. 2CMCh. 8.CM - Prob. 3CMCh. 8.CM - Prob. 4CMCh. 8.CM - Prob. 5CMCh. 8.CM - Prob. 6CMCh. 8.CM - Prob. 7CMCh. 8.CM - Prob. 8CMCh. 8.CM - Prob. 9CMCh. 8.CM - Prob. 10CMCh. 8.CM - Prob. 11CMCh. 8.CM - Prob. 12CMCh. 8.CM - Prob. 13CMCh. 8.CM - Prob. 14CMCh. 8.CM - Prob. 15CMCh. 8.CM - Prob. 16CMCh. 8.CM - Prob. 17CMCh. 8.CM - Prob. 18CMCh. 8.CM - Prob. 19CMCh. 8.CM - Prob. 20CMCh. 8.CM - Prob. 21CMCh. 8.CM - Prob. 22CMCh. 8.CM - Prob. 23CMCh. 8.CM - Prob. 24CMCh. 8.CM - Prob. 25CMCh. 8.CM - Prob. 26CMCh. 8.CM - Prob. 27CMCh. 8.CM - Prob. 28CMCh. 8.CM - Prob. 29CMCh. 8.CM - Prob. 30CMCh. 8.CM - Prob. 31CMCh. 8.CM - Prob. 32CMCh. 8.CM - Prob. 33CMCh. 8.CM - Prob. 34CMCh. 8.CM - Prob. 35CMCh. 8.CM - Prob. 36CMCh. 8.CM - Prob. 37CMCh. 8.CM - Prob. 38CMCh. 8.CM - Prob. 39CMCh. 8.CM - Prob. 40CMCh. 8.CM - Prob. 41CMCh. 8.CM - Prob. 42CMCh. 8.CM - Prob. 43CMCh. 8.CM - Prob. 44CMCh. 8.CM - Prob. 45CMCh. 8.CM - Prob. 46CMCh. 8.CM - Prob. 47CMCh. 8.CM - Prob. 48CMCh. 8.CM - Prob. 49CMCh. 8.CM - Prob. 50CMCh. 8.CM - Prob. 51CMCh. 8.CM - Prob. 52CMCh. 8.CM - Prob. 53CMCh. 8.CM - Prob. 54CMCh. 8.CM - Prob. 55CMCh. 8.CM - Prob. 56CMCh. 8.CM - For Problems 5564, solve inequality and express...Ch. 8.CM - Prob. 58CMCh. 8.CM - Prob. 59CMCh. 8.CM - Prob. 60CMCh. 8.CM - Prob. 61CMCh. 8.CM - Prob. 62CMCh. 8.CM - Prob. 63CMCh. 8.CM - Prob. 64CMCh. 8.CM - Prob. 65CMCh. 8.CM - For Problems 65-70, graph the following equations....Ch. 8.CM - Prob. 67CMCh. 8.CM - Prob. 68CMCh. 8.CM - Prob. 69CMCh. 8.CM - Prob. 70CMCh. 8.CM - Prob. 71CMCh. 8.CM - Prob. 72CMCh. 8.CM - Prob. 73CMCh. 8.CM - Prob. 74CMCh. 8.CM - Prob. 75CMCh. 8.CM - Prob. 76CMCh. 8.CM - Prob. 77CMCh. 8.CM - Prob. 78CMCh. 8.CM - Prob. 79CMCh. 8.CM - Prob. 80CMCh. 8.CM - Prob. 81CMCh. 8.CM - Prob. 82CMCh. 8.CM - Prob. 83CMCh. 8.CM - Prob. 84CMCh. 8.CM - Prob. 85CMCh. 8.CM - Prob. 86CMCh. 8.CM - Prob. 87CMCh. 8.CM - Prob. 88CM
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