 68.1: x2 + 4 = 0
 68.2: x3 + 4x2  21x = 0
 68.3: f(x) = 5x3 + 8x2  4x + 3
 68.4: r(x) = x5  x3  x + 1
 68.5: p(x) = x3 + 2x2  3x + 20
 68.6: f(x) = x3  4x2 + 6x  4
 68.7: v(x) = x3  3x2 + 4x  12
 68.8: f(x) = x3  3x2 + 9x + 13
 68.9: Write a polynomial function of least degree with integral coefficie...
 68.10: Write a polynomial function of least degree with integral coefficie...
 68.11: 3x + 8 = 0
 68.12: 2x2  5x + 12 = 0
 68.13: x3 + 9x = 0
 68.14: x4  81 = 0
 68.15: x4  16 = 0
 68.16: x5  8x3 + 16x = 0
 68.17: f(x) = x3  6x2 + 1
 68.18: g(x) = 5x3 + 8x2  4x + 3
 68.19: h(x) = 4x3  6x2 + 8x  5
 68.20: q(x) = x4 + 5x3 + 2x2  7x  9
 68.21: p(x) = x5  6x4  3x3 + 7x2  8x + 1
 68.22: f(x) = x10  x8 + x6  x4 + x2  1
 68.23: g(x) = x3 + 6x2 + 21x + 26
 68.24: h(x) = x3  6x2 + 10x  8
 68.25: f(x) = x3  5x2  7x + 51
 68.26: f(x) = x3  7x2 + 25x  175
 68.27: g(x) = 2x3  x2 + 28x + 51
 68.28: q(x) = 2x3  17x2 + 90x  41
 68.29: h(x) = 4x4 + 17x2 + 4
 68.30: p(x) = x4  9x3 + 24x2  6x  40
 68.31: r(x) = x4  6x3 + 12x2 + 6x  13
 68.32: h(x) = x4  15x3 + 70x2  70x  156
 68.33: 4, 1, 5
 68.34: 2, 2, 4, 6
 68.35: 4i, 3, 3
 68.36: 2i, 3i, 1
 68.37: 9, 1 + 2i
 68.38: 6, 2 + 2i
 68.39: How many positive real zeros, negative real zeros, and imaginary ze...
 68.40: Approximate all real zeros to the nearest tenth by graphing the fun...
 68.41: What is the meaning of the roots in this problem?
 68.42: Write an equation that represents the volume of the cylinder.
 68.43: What are the dimensions of the cylindrical part of the tank?
 68.44: Write a polynomial equation to model this situation.
 68.45: How much should he take from each dimension?
 68.46: OPEN ENDED Sketch the graph of a polynomial function that has the i...
 68.47: CHALLENGE If a sixthdegree polynomial equation has exactly five di...
 68.48: REASONING State the least degree a polynomial equation with real co...
 68.49: CHALLENGE Find a counterexample to disprove the following statement...
 68.50: Writing in Math Use the information about medication on page 362 to...
 68.51: ACT/SAT How many negative real zeros does f(x) = x5  2x4  4x3 + 4...
 68.52: REVIEW Tiles numbered from 1 to 6 are placed in a bag and are drawn...
 68.53: f(x) = x3  5x2 + 16x  7
 68.54: f(x) = x4 + 11x3  3x2 + 2x  5
 68.55: 15a2b2  5ab2c2
 68.56: 12p2  64p + 45
 68.57: 4y3 + 24y2 + 36y
 68.58: BASKETBALL In a recent season, Monique Currie of the Duke Blue Devi...
 68.59: a = {1, 5}; b = {1, 2}
 68.60: a = {1, 2}; b = {1, 2, 7, 14}
 68.61: a = {1, 3}; b = {1, 3, 9}
 68.62: a = {1, 2, 4}; b = {1, 2, 4, 8, 16}
Solutions for Chapter 68: Roots and Zeros
Full solutions for Algebra 2, Student Edition (MERRILL ALGEBRA 2)  1st Edition
ISBN: 9780078738302
Solutions for Chapter 68: Roots and Zeros
Get Full SolutionsSince 62 problems in chapter 68: Roots and Zeros have been answered, more than 60721 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Algebra 2, Student Edition (MERRILL ALGEBRA 2), edition: 1. Chapter 68: Roots and Zeros includes 62 full stepbystep solutions. Algebra 2, Student Edition (MERRILL ALGEBRA 2) was written by and is associated to the ISBN: 9780078738302. This expansive textbook survival guide covers the following chapters and their solutions.

Adjacency matrix of a graph.
Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected). Adjacency matrix of a graph. Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected).

Affine transformation
Tv = Av + Vo = linear transformation plus shift.

Cofactor Cij.
Remove row i and column j; multiply the determinant by (I)i + j •

Conjugate Gradient Method.
A sequence of steps (end of Chapter 9) to solve positive definite Ax = b by minimizing !x T Ax  x Tb over growing Krylov subspaces.

Dimension of vector space
dim(V) = number of vectors in any basis for V.

Dot product = Inner product x T y = XI Y 1 + ... + Xn Yn.
Complex dot product is x T Y . Perpendicular vectors have x T y = O. (AB)ij = (row i of A)T(column j of B).

Echelon matrix U.
The first nonzero entry (the pivot) in each row comes in a later column than the pivot in the previous row. All zero rows come last.

Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).
Use AT for complex A.

Graph G.
Set of n nodes connected pairwise by m edges. A complete graph has all n(n  1)/2 edges between nodes. A tree has only n  1 edges and no closed loops.

Hypercube matrix pl.
Row n + 1 counts corners, edges, faces, ... of a cube in Rn.

Identity matrix I (or In).
Diagonal entries = 1, offdiagonal entries = 0.

Indefinite matrix.
A symmetric matrix with eigenvalues of both signs (+ and  ).

Iterative method.
A sequence of steps intended to approach the desired solution.

Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).

Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , AjIb. Numerical methods approximate A I b by x j with residual b  Ax j in this subspace. A good basis for K j requires only multiplication by A at each step.

Pivot.
The diagonal entry (first nonzero) at the time when a row is used in elimination.

Rank one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.

Simplex method for linear programming.
The minimum cost vector x * is found by moving from comer to lower cost comer along the edges of the feasible set (where the constraints Ax = b and x > 0 are satisfied). Minimum cost at a comer!

Spanning set.
Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!

Special solutions to As = O.
One free variable is Si = 1, other free variables = o.