 3.1: In Exercises 14, graph the given quadratic function. Give eachfunct...
 3.2: In Exercises 14, graph the given quadratic function. Give eachfunct...
 3.3: In Exercises 14, graph the given quadratic function. Give eachfunct...
 3.4: In Exercises 14, graph the given quadratic function. Give eachfunct...
 3.5: In Exercises 513, find all zeros of each polynomial function.Then g...
 3.6: In Exercises 513, find all zeros of each polynomial function.Then g...
 3.7: In Exercises 513, find all zeros of each polynomial function.Then g...
 3.8: In Exercises 513, find all zeros of each polynomial function.Then g...
 3.9: In Exercises 513, find all zeros of each polynomial function.Then g...
 3.10: In Exercises 513, find all zeros of each polynomial function.Then g...
 3.11: In Exercises 513, find all zeros of each polynomial function.Then g...
 3.12: In Exercises 513, find all zeros of each polynomial function.Then g...
 3.13: In Exercises 513, find all zeros of each polynomial function.Then g...
 3.14: In Exercises 1419, solve each polynomial equation.x3  3x + 2 = 0
 3.15: In Exercises 1419, solve each polynomial equation.6x3  11x2 + 6x ...
 3.16: In Exercises 1419, solve each polynomial equation.(2x + 1)(3x  2)3...
 3.17: In Exercises 1419, solve each polynomial equation.2x3 + 5x2  200x ...
 3.18: In Exercises 1419, solve each polynomial equation.x4  x3  11x2 = ...
 3.19: In Exercises 1419, solve each polynomial equation.2x4 + x3  17x2 ...
 3.20: A company manufactures and sells bath cabinets. ThefunctionP(x) = ...
 3.21: Among all pairs of numbers whose sum is 18, find a pairwhose produ...
 3.22: The base of a triangle measures 40 inches minus twice themeasure of...
 3.23: In Exercises 2324, divide, using synthetic division if possible.(6x...
 3.24: In Exercises 2324, divide, using synthetic division if possible.(2x...
 3.25: In Exercises 2526, find an nthdegree polynomial function withreal ...
 3.26: In Exercises 2526, find an nthdegree polynomial function withreal ...
 3.27: Does f(x) = x3  x  5 have a real zero between 1 and 2?
 3.28: In Exercises 2729, divide using long division.(10x3  26x2 + 17x  ...
 3.29: In Exercises 2729, divide using long division.(4x4 + 6x3 + 3x  1) ...
 3.30: In Exercises 3031, divide using synthetic division.(3x4 + 11x3  20...
 3.31: In Exercises 3031, divide using synthetic division.(3x4  2x2  10x...
 3.32: Given f(x) = 2x3  7x2 + 9x  3, use the RemainderTheorem to find f...
 3.33: Use synthetic division to divide f(x) = 2x3 + x2  13x + 6by x  2....
 3.34: Solve the equation x3  17x + 4 = 0 given that 4 is aroot.
 3.35: In Exercises 3536, use the Rational Zero Theorem to list allpossibl...
 3.36: In Exercises 3536, use the Rational Zero Theorem to list allpossibl...
 3.37: In Exercises 3738, use Descartess Rule of Signs to determine thepos...
 3.38: In Exercises 3738, use Descartess Rule of Signs to determine thepos...
 3.39: Use Descartess Rule of Signs to explain why2x4 + 6x2 + 8 = 0 has no...
 3.40: For Exercises 4046,a. List all possible rational roots or rational ...
 3.41: For Exercises 4046,a. List all possible rational roots or rational ...
 3.42: For Exercises 4046,a. List all possible rational roots or rational ...
 3.43: For Exercises 4046,a. List all possible rational roots or rational ...
 3.44: For Exercises 4046,a. List all possible rational roots or rational ...
 3.45: For Exercises 4046,a. List all possible rational roots or rational ...
 3.46: For Exercises 4046,a. List all possible rational roots or rational ...
 3.47: In Exercises 4748, find an nthdegree polynomial function withreal ...
 3.48: In Exercises 4748, find an nthdegree polynomial function withreal ...
 3.49: In Exercises 4950, find all the zeros of each polynomial functionan...
 3.50: In Exercises 4950, find all the zeros of each polynomial functionan...
 3.51: In Exercises 5154, graphs of fifthdegree polynomial functionsare s...
 3.52: In Exercises 5154, graphs of fifthdegree polynomial functionsare s...
 3.53: In Exercises 5154, graphs of fifthdegree polynomial functionsare s...
 3.54: In Exercises 5154, graphs of fifthdegree polynomial functionsare s...
 3.55: In Exercises 5556, use transformations of f(x) = 1x or f(x) = 1x2to...
 3.56: In Exercises 5556, use transformations of f(x) = 1x or f(x) = 1x2to...
 3.57: In Exercises 5764, find the vertical asymptotes, if any, thehorizon...
 3.58: In Exercises 5764, find the vertical asymptotes, if any, thehorizon...
 3.59: In Exercises 5764, find the vertical asymptotes, if any, thehorizon...
 3.60: In Exercises 5764, find the vertical asymptotes, if any, thehorizon...
 3.61: In Exercises 5764, find the vertical asymptotes, if any, thehorizon...
 3.62: In Exercises 5764, find the vertical asymptotes, if any, thehorizon...
 3.63: In Exercises 5764, find the vertical asymptotes, if any, thehorizon...
 3.64: In Exercises 5764, find the vertical asymptotes, if any, thehorizon...
 3.65: A company is planning to manufacture affordable graphingcalculators...
 3.66: Exercises 6667 involve rational functions that model thegiven situa...
 3.67: Exercises 6667 involve rational functions that model thegiven situa...
 3.68: The bar graph shows the population of the United States, inmillions...
 3.69: In Exercises 6974, solve each inequality and graph the solutionset ...
 3.70: In Exercises 6974, solve each inequality and graph the solutionset ...
 3.71: In Exercises 6974, solve each inequality and graph the solutionset ...
 3.72: In Exercises 6974, solve each inequality and graph the solutionset ...
 3.73: In Exercises 6974, solve each inequality and graph the solutionset ...
 3.74: In Exercises 6974, solve each inequality and graph the solutionset ...
 3.75: The graph shows stopping distances for motorcycles atvarious speeds...
 3.76: Use the position functions(t) = 16t2 + v0t + s0to solve this probl...
 3.77: Solve the variation problems in Exercises 7782.Many areas of Northe...
 3.78: Solve the variation problems in Exercises 7782.The distance that a ...
 3.79: Solve the variation problems in Exercises 7782.The pitch of a music...
 3.80: Solve the variation problems in Exercises 7782.The loudness of a st...
 3.81: Solve the variation problems in Exercises 7782.The time required to...
 3.82: Solve the variation problems in Exercises 7782.The volume of a pyra...
 3.83: Heart rates and life spans of most mammals can be modeledusing inve...
Solutions for Chapter 3: Polynomial and Rational Functions
Full solutions for College Algebra  7th Edition
ISBN: 9780134469164
Solutions for Chapter 3: Polynomial and Rational Functions
Get Full SolutionsChapter 3: Polynomial and Rational Functions includes 83 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: College Algebra , edition: 7. Since 83 problems in chapter 3: Polynomial and Rational Functions have been answered, more than 29774 students have viewed full stepbystep solutions from this chapter. College Algebra was written by and is associated to the ISBN: 9780134469164.

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

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).

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

Ellipse (or ellipsoid) x T Ax = 1.
A must be positive definite; the axes of the ellipse are eigenvectors of A, with lengths 1/.JI. (For IIx II = 1 the vectors y = Ax lie on the ellipse IIA1 yll2 = Y T(AAT)1 Y = 1 displayed by eigshow; axis lengths ad

Fourier matrix F.
Entries Fjk = e21Cijk/n give orthogonal columns FT F = nI. Then y = Fe is the (inverse) Discrete Fourier Transform Y j = L cke21Cijk/n.

Full column rank r = n.
Independent columns, N(A) = {O}, no free variables.

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.

Independent vectors VI, .. " vk.
No combination cl VI + ... + qVk = zero vector unless all ci = O. If the v's are the columns of A, the only solution to Ax = 0 is x = o.

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.

Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = O.

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

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

Orthogonal subspaces.
Every v in V is orthogonal to every w in W.

Pivot columns of A.
Columns that contain pivots after row reduction. These are not combinations of earlier columns. The pivot columns are a basis for the column space.

Projection matrix P onto subspace S.
Projection p = P b is the closest point to b in S, error e = b  Pb is perpendicularto S. p 2 = P = pT, eigenvalues are 1 or 0, eigenvectors are in S or S...L. If columns of A = basis for S then P = A (AT A) 1 AT.

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.

Row picture of Ax = b.
Each equation gives a plane in Rn; the planes intersect at x.

Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).

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

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·