 2.3.2.3.1: Two forms of the Division Algorithm are shown below. Identify and l...
 2.3.2.3.2: In Exercises 2 7, fill in the blanks.
 2.3.2.3.3: In Exercises 2 7, fill in the blanks.
 2.3.2.3.4: In Exercises 2 7, fill in the blanks.
 2.3.2.3.5: In Exercises 2 7, fill in the blanks.
 2.3.2.3.6: In Exercises 2 7, fill in the blanks.
 2.3.2.3.7: In Exercises 2 7, fill in the blanks.
 2.3.2.3.8: In Exercises 114, use long division to divide.
 2.3.2.3.9: In Exercises 114, use long division to divide.
 2.3.2.3.10: In Exercises 114, use long division to divide.
 2.3.2.3.11: In Exercises 114, use long division to divide.
 2.3.2.3.12: In Exercises 114, use long division to divide.
 2.3.2.3.13: In Exercises 114, use long division to divide.
 2.3.2.3.14: In Exercises 114, use long division to divide.
 2.3.2.3.15: In Exercises 1524, use synthetic division to divide.
 2.3.2.3.16: In Exercises 1524, use synthetic division to divide.
 2.3.2.3.17: In Exercises 1524, use synthetic division to divide.
 2.3.2.3.18: In Exercises 1524, use synthetic division to divide.
 2.3.2.3.19: In Exercises 1524, use synthetic division to divide.
 2.3.2.3.20: In Exercises 1524, use synthetic division to divide.
 2.3.2.3.21: In Exercises 1524, use synthetic division to divide.
 2.3.2.3.22: In Exercises 1524, use synthetic division to divide.
 2.3.2.3.23: In Exercises 1524, use synthetic division to divide.
 2.3.2.3.24: In Exercises 1524, use synthetic division to divide.
 2.3.2.3.25: Graphical Analysis In Exercises 2528, use a graphing utility to gra...
 2.3.2.3.26: Graphical Analysis In Exercises 2528, use a graphing utility to gra...
 2.3.2.3.27: Graphical Analysis In Exercises 2528, use a graphing utility to gra...
 2.3.2.3.28: Graphical Analysis In Exercises 2528, use a graphing utility to gra...
 2.3.2.3.29: In Exercises 2934, write the function in the form for the given val...
 2.3.2.3.30: In Exercises 2934, write the function in the form for the given val...
 2.3.2.3.31: In Exercises 2934, write the function in the form for the given val...
 2.3.2.3.32: In Exercises 2934, write the function in the form for the given val...
 2.3.2.3.33: In Exercises 2934, write the function in the form for the given val...
 2.3.2.3.34: In Exercises 2934, write the function in the form for the given val...
 2.3.2.3.35: In Exercises 3538, use the Remainder Theorem and synthetic division...
 2.3.2.3.36: In Exercises 3538, use the Remainder Theorem and synthetic division...
 2.3.2.3.37: In Exercises 3538, use the Remainder Theorem and synthetic division...
 2.3.2.3.38: In Exercises 3538, use the Remainder Theorem and synthetic division...
 2.3.2.3.39: In Exercises 39 42, use synthetic division to show that is a soluti...
 2.3.2.3.40: In Exercises 39 42, use synthetic division to show that is a soluti...
 2.3.2.3.41: In Exercises 39 42, use synthetic division to show that is a soluti...
 2.3.2.3.42: In Exercises 39 42, use synthetic division to show that is a soluti...
 2.3.2.3.43: In Exercises 43 48, (a) verify the given factors of the function (b...
 2.3.2.3.44: In Exercises 43 48, (a) verify the given factors of the function (b...
 2.3.2.3.45: In Exercises 43 48, (a) verify the given factors of the function (b...
 2.3.2.3.46: In Exercises 43 48, (a) verify the given factors of the function (b...
 2.3.2.3.47: In Exercises 43 48, (a) verify the given factors of the function (b...
 2.3.2.3.48: In Exercises 43 48, (a) verify the given factors of the function (b...
 2.3.2.3.49: In Exercises 49 52, use the Rational Zero Test to list all possible...
 2.3.2.3.50: In Exercises 49 52, use the Rational Zero Test to list all possible...
 2.3.2.3.51: In Exercises 49 52, use the Rational Zero Test to list all possible...
 2.3.2.3.52: In Exercises 49 52, use the Rational Zero Test to list all possible...
 2.3.2.3.53: In Exercises 5360, find all real zeros of the polynomial function.
 2.3.2.3.54: In Exercises 5360, find all real zeros of the polynomial function.
 2.3.2.3.55: In Exercises 5360, find all real zeros of the polynomial function.
 2.3.2.3.56: In Exercises 5360, find all real zeros of the polynomial function.
 2.3.2.3.57: In Exercises 5360, find all real zeros of the polynomial function.
 2.3.2.3.58: In Exercises 5360, find all real zeros of the polynomial function.
 2.3.2.3.59: In Exercises 5360, find all real zeros of the polynomial function.
 2.3.2.3.60: In Exercises 5360, find all real zeros of the polynomial function.
 2.3.2.3.61: Graphical Analysis In Exercises 6164, (a) use the zero or root feat...
 2.3.2.3.62: Graphical Analysis In Exercises 6164, (a) use the zero or root feat...
 2.3.2.3.63: Graphical Analysis In Exercises 6164, (a) use the zero or root feat...
 2.3.2.3.64: Graphical Analysis In Exercises 6164, (a) use the zero or root feat...
 2.3.2.3.65: In Exercises 6568, use Descartess Rule of Signs to determine the po...
 2.3.2.3.66: In Exercises 6568, use Descartess Rule of Signs to determine the po...
 2.3.2.3.67: In Exercises 6568, use Descartess Rule of Signs to determine the po...
 2.3.2.3.68: In Exercises 6568, use Descartess Rule of Signs to determine the po...
 2.3.2.3.69: In Exercises 6974, (a) use Descartess Rule of Signs to determine th...
 2.3.2.3.70: In Exercises 6974, (a) use Descartess Rule of Signs to determine th...
 2.3.2.3.71: In Exercises 6974, (a) use Descartess Rule of Signs to determine th...
 2.3.2.3.72: In Exercises 6974, (a) use Descartess Rule of Signs to determine th...
 2.3.2.3.73: In Exercises 6974, (a) use Descartess Rule of Signs to determine th...
 2.3.2.3.74: In Exercises 6974, (a) use Descartess Rule of Signs to determine th...
 2.3.2.3.75: In Exercises 7578, use synthetic division to verify the upper and l...
 2.3.2.3.76: In Exercises 7578, use synthetic division to verify the upper and l...
 2.3.2.3.77: In Exercises 7578, use synthetic division to verify the upper and l...
 2.3.2.3.78: In Exercises 7578, use synthetic division to verify the upper and l...
 2.3.2.3.79: In Exercises 7982, find the rational zeros of the polynomial function.
 2.3.2.3.80: In Exercises 7982, find the rational zeros of the polynomial function.
 2.3.2.3.81: In Exercises 7982, find the rational zeros of the polynomial function.
 2.3.2.3.82: In Exercises 7982, find the rational zeros of the polynomial function.
 2.3.2.3.83: In Exercises 83 86, match the cubic function with the correct numbe...
 2.3.2.3.84: In Exercises 83 86, match the cubic function with the correct numbe...
 2.3.2.3.85: In Exercises 83 86, match the cubic function with the correct numbe...
 2.3.2.3.86: In Exercises 83 86, match the cubic function with the correct numbe...
 2.3.2.3.87: In Exercises 87 90, the graph of is shown. Use the graph as an aid ...
 2.3.2.3.88: In Exercises 87 90, the graph of is shown. Use the graph as an aid ...
 2.3.2.3.89: In Exercises 87 90, the graph of is shown. Use the graph as an aid ...
 2.3.2.3.90: In Exercises 87 90, the graph of is shown. Use the graph as an aid ...
 2.3.2.3.91: U.S. Population The table shows the populations of the United State...
 2.3.2.3.92: Energy The number of coal mines C in the United States from 1980 to...
 2.3.2.3.93: Geometry A rectangular package sent by a delivery service can have ...
 2.3.2.3.94: Automobile Emissions The number of parts per million of nitric oxid...
 2.3.2.3.95: True or False? In Exercises 95 and 96, determine whether the statem...
 2.3.2.3.96: True or False? In Exercises 95 and 96, determine whether the statem...
 2.3.2.3.97: Think About It In Exercises 97 and 98, the graph of a cubic polynom...
 2.3.2.3.98: Think About It In Exercises 97 and 98, the graph of a cubic polynom...
 2.3.2.3.99: Think About It Let be a quartic polynomial with leading coefficient...
 2.3.2.3.100: Think About It Let be a cubic polynomial with leading coefficient a...
 2.3.2.3.101: Think About It Find the value of k such that is a factor of x3 kx2 ...
 2.3.2.3.102: Think About It Find the value of k such that is a factor of x3 kx2 ...
 2.3.2.3.103: Writing Complete each polynomial division. Write a brief descriptio...
 2.3.2.3.104: Writing Write a short paragraph explaining how you can check polyno...
 2.3.2.3.105: In Exercises 105108, use any convenient method to solve the quadrat...
 2.3.2.3.106: In Exercises 105108, use any convenient method to solve the quadrat...
 2.3.2.3.107: In Exercises 105108, use any convenient method to solve the quadrat...
 2.3.2.3.108: In Exercises 105108, use any convenient method to solve the quadrat...
 2.3.2.3.109: In Exercises 109112, find a polynomial function that has the given ...
 2.3.2.3.110: In Exercises 109112, find a polynomial function that has the given ...
 2.3.2.3.111: In Exercises 109112, find a polynomial function that has the given ...
 2.3.2.3.112: In Exercises 109112, find a polynomial function that has the given ...
Solutions for Chapter 2.3: Real Zeros of Polynomial Functions
Full solutions for Precalculus With Limits A Graphing Approach  5th Edition
ISBN: 9780618851522
Solutions for Chapter 2.3: Real Zeros of Polynomial Functions
Get Full SolutionsThis textbook survival guide was created for the textbook: Precalculus With Limits A Graphing Approach, edition: 5. This expansive textbook survival guide covers the following chapters and their solutions. Precalculus With Limits A Graphing Approach was written by and is associated to the ISBN: 9780618851522. Chapter 2.3: Real Zeros of Polynomial Functions includes 112 full stepbystep solutions. Since 112 problems in chapter 2.3: Real Zeros of Polynomial Functions have been answered, more than 36196 students have viewed full stepbystep solutions from this chapter.

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

Cholesky factorization
A = CTC = (L.J]))(L.J]))T for positive definite A.

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

Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.

Companion matrix.
Put CI, ... ,Cn in row n and put n  1 ones just above the main diagonal. Then det(A  AI) = ±(CI + c2A + C3A 2 + .•. + cnA nl  An).

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.

Free variable Xi.
Column i has no pivot in elimination. We can give the n  r free variables any values, then Ax = b determines the r pivot variables (if solvable!).

GramSchmidt orthogonalization A = QR.
Independent columns in A, orthonormal columns in Q. Each column q j of Q is a combination of the first j columns of A (and conversely, so R is upper triangular). Convention: diag(R) > o.

Hilbert matrix hilb(n).
Entries HU = 1/(i + j 1) = Jd X i 1 xj1dx. Positive definite but extremely small Amin and large condition number: H is illconditioned.

Matrix multiplication AB.
The i, j entry of AB is (row i of A)·(column j of B) = L aikbkj. By columns: Column j of AB = A times column j of B. By rows: row i of A multiplies B. Columns times rows: AB = sum of (column k)(row k). All these equivalent definitions come from the rule that A B times x equals A times B x .

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

Norm
IIA II. The ".e 2 norm" of A is the maximum ratio II Ax II/l1x II = O"max· Then II Ax II < IIAllllxll and IIABII < IIAIIIIBII and IIA + BII < IIAII + IIBII. Frobenius norm IIAII} = L La~. The.e 1 and.e oo norms are largest column and row sums of laij I.

Normal equation AT Ax = ATb.
Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)·(b  Ax) = o.

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

Orthonormal vectors q 1 , ... , q n·
Dot products are q T q j = 0 if i =1= j and q T q i = 1. The matrix Q with these orthonormal columns has Q T Q = I. If m = n then Q T = Q 1 and q 1 ' ... , q n is an orthonormal basis for Rn : every v = L (v T q j )q j •

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.

Skewsymmetric matrix K.
The transpose is K, since Kij = Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.

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

Triangle inequality II u + v II < II u II + II v II.
For matrix norms II A + B II < II A II + II B II·

Vector addition.
v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.