 4.3.1: Exer. 16: Find a polynomial f(x) of degree 3 that has the indicated...
 4.3.2: Exer. 16: Find a polynomial f(x) of degree 3 that has the indicated...
 4.3.3: Exer. 16: Find a polynomial f(x) of degree 3 that has the indicated...
 4.3.4: Exer. 16: Find a polynomial f(x) of degree 3 that has the indicated...
 4.3.5: Exer. 16: Find a polynomial f(x) of degree 3 that has the indicated...
 4.3.6: Exer. 16: Find a polynomial f(x) of degree 3 that has the indicated...
 4.3.7: Find a polynomial of degree 4 with leading coefficient 1 such that ...
 4.3.8: Find a polynomial of degree 4 with leading coefficient 1 such that ...
 4.3.9: Find a polynomial of degree 6 such that 0 and 3 are both zeros of m...
 4.3.10: Find a polynomial of degree 7 such that and 2 are both zeros of mul...
 4.3.11: Find the thirddegree polynomial function whose graph is shown in t...
 4.3.12: Find the fourthdegree polynomial function whose graph is shown in ...
 4.3.13: Exer. 1314: Find the polynomial function of degree 3 whose graph is...
 4.3.14: Exer. 1314: Find the polynomial function of degree 3 whose graph is...
 4.3.15: Exer. 1522: Find the zeros of f(x), and state the multiplicity of e...
 4.3.16: Exer. 1522: Find the zeros of f(x), and state the multiplicity of e...
 4.3.17: Exer. 1522: Find the zeros of f(x), and state the multiplicity of e...
 4.3.18: Exer. 1522: Find the zeros of f(x), and state the multiplicity of e...
 4.3.19: Exer. 1522: Find the zeros of f(x), and state the multiplicity of e...
 4.3.20: Exer. 1522: Find the zeros of f(x), and state the multiplicity of e...
 4.3.21: Exer. 1522: Find the zeros of f(x), and state the multiplicity of e...
 4.3.22: Exer. 1522: Find the zeros of f(x), and state the multiplicity of e...
 4.3.23: Exer. 2326: Show that the number is a zero of f(x) of the given mul...
 4.3.24: Exer. 2326: Show that the number is a zero of f(x) of the given mul...
 4.3.25: Exer. 2326: Show that the number is a zero of f(x) of the given mul...
 4.3.26: Exer. 2326: Show that the number is a zero of f(x) of the given mul...
 4.3.27: Exer. 2734: Use Descartes rule of signs to determine the number of ...
 4.3.28: Exer. 2734: Use Descartes rule of signs to determine the number of ...
 4.3.29: Exer. 2734: Use Descartes rule of signs to determine the number of ...
 4.3.30: Exer. 2734: Use Descartes rule of signs to determine the number of ...
 4.3.31: Exer. 2734: Use Descartes rule of signs to determine the number of ...
 4.3.32: Exer. 2734: Use Descartes rule of signs to determine the number of ...
 4.3.33: Exer. 2734: Use Descartes rule of signs to determine the number of ...
 4.3.34: Exer. 2734: Use Descartes rule of signs to determine the number of ...
 4.3.35: Exer. 3540: Applying the theorem on bounds for real zeros of polyno...
 4.3.36: Exer. 3540: Applying the theorem on bounds for real zeros of polyno...
 4.3.37: Exer. 3540: Applying the theorem on bounds for real zeros of polyno...
 4.3.38: Exer. 3540: Applying the theorem on bounds for real zeros of polyno...
 4.3.39: Exer. 3540: Applying the theorem on bounds for real zeros of polyno...
 4.3.40: Exer. 3540: Applying the theorem on bounds for real zeros of polyno...
 4.3.41: Exer. 4142: Find a factored form for a polynomial function f that h...
 4.3.42: Exer. 4142: Find a factored form for a polynomial function f that h...
 4.3.43: Exer. 4344: (a) Find a factored form for a polynomial function f th...
 4.3.44: Exer. 4344: (a) Find a factored form for a polynomial function f th...
 4.3.45: Exer. 4548: Is there a polynomial of the given degree n whose graph...
 4.3.46: Exer. 4548: Is there a polynomial of the given degree n whose graph...
 4.3.47: Exer. 4548: Is there a polynomial of the given degree n whose graph...
 4.3.48: Exer. 4548: Is there a polynomial of the given degree n whose graph...
 4.3.49: A scientist has limited data on the temperature T (in C) during a 2...
 4.3.50: A polynomial of degree 3 with zeros at , , and and with for is a th...
Solutions for Chapter 4.3: Zeros of Polynomials
Full solutions for Algebra and Trigonometry with Analytic Geometry  12th Edition
ISBN: 9780495559719
Solutions for Chapter 4.3: Zeros of Polynomials
Get Full SolutionsSince 50 problems in chapter 4.3: Zeros of Polynomials have been answered, more than 35293 students have viewed full stepbystep solutions from this chapter. Algebra and Trigonometry with Analytic Geometry was written by and is associated to the ISBN: 9780495559719. This textbook survival guide was created for the textbook: Algebra and Trigonometry with Analytic Geometry, edition: 12. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 4.3: Zeros of Polynomials includes 50 full stepbystep solutions.

Back substitution.
Upper triangular systems are solved in reverse order Xn to Xl.

CayleyHamilton Theorem.
peA) = det(A  AI) has peA) = zero matrix.

Cross product u xv in R3:
Vector perpendicular to u and v, length Ilullllvlll sin el = area of parallelogram, u x v = "determinant" of [i j k; UI U2 U3; VI V2 V3].

Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then SI AS = A = eigenvalue matrix.

Diagonalization
A = S1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k SI.

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

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.

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

GaussJordan method.
Invert A by row operations on [A I] to reach [I AI].

Incidence matrix of a directed graph.
The m by n edgenode incidence matrix has a row for each edge (node i to node j), with entries 1 and 1 in columns i and j .

Kirchhoff's Laws.
Current Law: net current (in minus out) is zero at each node. Voltage Law: Potential differences (voltage drops) add to zero around any closed loop.

Multiplication Ax
= Xl (column 1) + ... + xn(column n) = combination of columns.

Nullspace N (A)
= All solutions to Ax = O. Dimension n  r = (# columns)  rank.

Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.

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

Right inverse A+.
If A has full row rank m, then A+ = AT(AAT)l has AA+ = 1m.

Rotation matrix
R = [~ CS ] rotates the plane by () and R 1 = RT rotates back by (). Eigenvalues are eiO and eiO , eigenvectors are (1, ±i). c, s = cos (), sin ().

Spectrum of A = the set of eigenvalues {A I, ... , An}.
Spectral radius = max of IAi I.

Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.

Symmetric matrix A.
The transpose is AT = A, and aU = a ji. AI is also symmetric.