 5.6.1: Find the sum and the product of the complex numbers and 3 + 5i 3 ...
 5.6.2: In the complex number system, find the complex solutions of the equ...
 5.6.3: Every polynomial function of odd degree with real coefficients will...
 5.6.4: If is a zero of a polynomial function of degree 5 with real coeffic...
 5.6.5: True or False A polynomial function of degree n with real coefficie...
 5.6.6: True or False A polynomial function of degree 4 with real coefficie...
 5.6.7: In 716, information is given about a polynomial function f(x) whose...
 5.6.8: In 716, information is given about a polynomial function f(x) whose...
 5.6.9: In 716, information is given about a polynomial function f(x) whose...
 5.6.10: In 716, information is given about a polynomial function f(x) whose...
 5.6.11: In 716, information is given about a polynomial function f(x) whose...
 5.6.12: In 716, information is given about a polynomial function f(x) whose...
 5.6.13: In 716, information is given about a polynomial function f(x) whose...
 5.6.14: In 716, information is given about a polynomial function f(x) whose...
 5.6.15: In 716, information is given about a polynomial function f(x) whose...
 5.6.16: In 716, information is given about a polynomial function f(x) whose...
 5.6.17: In 1722, form a polynomial function f(x) with real coefficients hav...
 5.6.18: In 1722, form a polynomial function f(x) with real coefficients hav...
 5.6.19: In 1722, form a polynomial function f(x) with real coefficients hav...
 5.6.20: In 1722, form a polynomial function f(x) with real coefficients hav...
 5.6.21: In 1722, form a polynomial function f(x) with real coefficients hav...
 5.6.22: In 1722, form a polynomial function f(x) with real coefficients hav...
 5.6.23: In 2330, use the given zero to find the remaining zeros of each fun...
 5.6.24: In 2330, use the given zero to find the remaining zeros of each fun...
 5.6.25: In 2330, use the given zero to find the remaining zeros of each fun...
 5.6.26: In 2330, use the given zero to find the remaining zeros of each fun...
 5.6.27: In 2330, use the given zero to find the remaining zeros of each fun...
 5.6.28: In 2330, use the given zero to find the remaining zeros of each fun...
 5.6.29: In 2330, use the given zero to find the remaining zeros of each fun...
 5.6.30: In 2330, use the given zero to find the remaining zeros of each fun...
 5.6.31: In 3140, find the complex zeros of each polynomial function.Write f...
 5.6.32: In 3140, find the complex zeros of each polynomial function.Write f...
 5.6.33: In 3140, find the complex zeros of each polynomial function.Write f...
 5.6.34: In 3140, find the complex zeros of each polynomial function.Write f...
 5.6.35: In 3140, find the complex zeros of each polynomial function.Write f...
 5.6.36: In 3140, find the complex zeros of each polynomial function.Write f...
 5.6.37: In 3140, find the complex zeros of each polynomial function.Write f...
 5.6.38: In 3140, find the complex zeros of each polynomial function.Write f...
 5.6.39: In 3140, find the complex zeros of each polynomial function.Write f...
 5.6.40: In 3140, find the complex zeros of each polynomial function.Write f...
 5.6.41: f1x2 is a polynomial function of degree 3 whose coefficients are re...
 5.6.42: f1x2 is a polynomial function of degree 3 whose coefficients are re...
 5.6.43: f1x2 is a polynomial function of degree 4 whose coefficients are re...
 5.6.44: f1x2 is a polynomial function of degree 4 whose coefficients are re...
Solutions for Chapter 5.6: Algebra and Trigonometry 9th Edition
Full solutions for Algebra and Trigonometry  9th Edition
ISBN: 9780321716569
Solutions for Chapter 5.6
Get Full SolutionsThis textbook survival guide was created for the textbook: Algebra and Trigonometry, edition: 9. Chapter 5.6 includes 44 full stepbystep solutions. Since 44 problems in chapter 5.6 have been answered, more than 57891 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. Algebra and Trigonometry was written by and is associated to the ISBN: 9780321716569.

Basis for V.
Independent vectors VI, ... , v d whose linear combinations give each vector in V as v = CIVI + ... + CdVd. V has many bases, each basis gives unique c's. A vector space has many bases!

Column picture of Ax = b.
The vector b becomes a combination of the columns of A. The system is solvable only when b is in the column space C (A).

Eigenvalue A and eigenvector x.
Ax = AX with x#O so det(A  AI) = o.

Elimination matrix = Elementary matrix Eij.
The identity matrix with an extra eij in the i, j entry (i # j). Then Eij A subtracts eij times row j of A from row i.

Fundamental Theorem.
The nullspace N (A) and row space C (AT) are orthogonal complements in Rn(perpendicular from Ax = 0 with dimensions rand n  r). Applied to AT, the column space C(A) is the orthogonal complement of N(AT) in Rm.

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

Jordan form 1 = M 1 AM.
If A has s independent eigenvectors, its "generalized" eigenvector matrix M gives 1 = diag(lt, ... , 1s). The block his Akh +Nk where Nk has 1 's on diagonall. Each block has one eigenvalue Ak and one eigenvector.

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

Lucas numbers
Ln = 2,J, 3, 4, ... satisfy Ln = L n l +Ln 2 = A1 +A~, with AI, A2 = (1 ± /5)/2 from the Fibonacci matrix U~]' Compare Lo = 2 with Fo = O.

Multiplier eij.
The pivot row j is multiplied by eij and subtracted from row i to eliminate the i, j entry: eij = (entry to eliminate) / (jth pivot).

Nilpotent matrix N.
Some power of N is the zero matrix, N k = o. The only eigenvalue is A = 0 (repeated n times). Examples: triangular matrices with zero diagonal.

Nullspace matrix N.
The columns of N are the n  r special solutions to As = O.

Partial pivoting.
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.

Pseudoinverse A+ (MoorePenrose inverse).
The n by m matrix that "inverts" A from column space back to row space, with N(A+) = N(AT). A+ A and AA+ are the projection matrices onto the row space and column space. Rank(A +) = rank(A).

Reduced row echelon form R = rref(A).
Pivots = 1; zeros above and below pivots; the r nonzero rows of R give a basis for the row space of A.

Saddle point of I(x}, ... ,xn ).
A point where the first derivatives of I are zero and the second derivative matrix (a2 II aXi ax j = Hessian matrix) is indefinite.

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!

Solvable system Ax = b.
The right side b is in the column space of A.

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

Toeplitz matrix.
Constant down each diagonal = timeinvariant (shiftinvariant) filter.