 3.1: In Exercises 1 and 2, use synthetic division to divide the first po...
 3.2: In Exercises 1 and 2, use synthetic division to divide the first po...
 3.3: In Exercises 3 to 6, use the Remainder Theorem to find P(c).
 3.4: In Exercises 3 to 6, use the Remainder Theorem to find P(c).
 3.5: In Exercises 3 to 6, use the Remainder Theorem to find P(c).
 3.6: In Exercises 3 to 6, use the Remainder Theorem to find P(c).
 3.7: In Exercises 7 to 10, use synthetic division to show that c is a ze...
 3.8: In Exercises 7 to 10, use synthetic division to show that c is a ze...
 3.9: In Exercises 7 to 10, use synthetic division to show that c is a ze...
 3.10: In Exercises 7 to 10, use synthetic division to show that c is a ze...
 3.11: In Exercises 11 and 12, use the Factor Theorem to determine whether...
 3.12: In Exercises 11 and 12, use the Factor Theorem to determine whether...
 3.13: In Exercises 13 and 14, determine the farleft and the farright be...
 3.14: In Exercises 13 and 14, determine the farleft and the farright be...
 3.15: In Exercises 15 and 16, use the maximum and minimum features of a g...
 3.16: In Exercises 15 and 16, use the maximum and minimum features of a g...
 3.17: In Exercises 17 and 18, use the Intermediate Value Theorem to verif...
 3.18: In Exercises 17 and 18, use the Intermediate Value Theorem to verif...
 3.19: In Exercises 19 and 20, determine the xintercepts of the graph of ...
 3.20: In Exercises 19 and 20, determine the xintercepts of the graph of ...
 3.21: In Exercises 21 to 26, graph the polynomial function.
 3.22: In Exercises 21 to 26, graph the polynomial function.
 3.23: In Exercises 21 to 26, graph the polynomial function.
 3.24: In Exercises 21 to 26, graph the polynomial function.
 3.25: In Exercises 21 to 26, graph the polynomial function.
 3.26: In Exercises 21 to 26, graph the polynomial function.
 3.27: In Exercises 27 to 32, use the Rational Zero Theorem to list all po...
 3.28: In Exercises 27 to 32, use the Rational Zero Theorem to list all po...
 3.29: In Exercises 27 to 32, use the Rational Zero Theorem to list all po...
 3.30: In Exercises 27 to 32, use the Rational Zero Theorem to list all po...
 3.31: In Exercises 27 to 32, use the Rational Zero Theorem to list all po...
 3.32: In Exercises 27 to 32, use the Rational Zero Theorem to list all po...
 3.33: In Exercises 33 to 36, use Descartes Rule of Signs to state the num...
 3.34: In Exercises 33 to 36, use Descartes Rule of Signs to state the num...
 3.35: In Exercises 33 to 36, use Descartes Rule of Signs to state the num...
 3.36: In Exercises 33 to 36, use Descartes Rule of Signs to state the num...
 3.37: In Exercises 37 to 42, find the zeros of the polynomial function.
 3.38: In Exercises 37 to 42, find the zeros of the polynomial function.
 3.39: In Exercises 37 to 42, find the zeros of the polynomial function.
 3.40: In Exercises 37 to 42, find the zeros of the polynomial function.
 3.41: In Exercises 37 to 42, find the zeros of the polynomial function.
 3.42: In Exercises 37 to 42, find the zeros of the polynomial function.
 3.43: In Exercises 43 and 44, find all the zeros of P and write P as a pr...
 3.44: In Exercises 43 and 44, find all the zeros of P and write P as a pr...
 3.45: In Exercises 45 and 46, use the given zero to find the remaining ze...
 3.46: In Exercises 45 and 46, use the given zero to find the remaining ze...
 3.47: In Exercises 47 to 50, find the requested polynomial function.
 3.48: In Exercises 47 to 50, find the requested polynomial function.
 3.49: In Exercises 47 to 50, find the requested polynomial function.
 3.50: In Exercises 47 to 50, find the requested polynomial function.
 3.51: In Exercises 51 and 52, determine the domain of the rational function.
 3.52: In Exercises 51 and 52, determine the domain of the rational function.
 3.53: In Exercises 53 and 54, determine the vertical asymptotes for the g...
 3.54: In Exercises 53 and 54, determine the vertical asymptotes for the g...
 3.55: In Exercises 55 and 56, determine the horizontal asymptote for the ...
 3.56: In Exercises 55 and 56, determine the horizontal asymptote for the ...
 3.57: In Exercises 57 and 58, determine the slant asymptote for the graph...
 3.58: In Exercises 57 and 58, determine the slant asymptote for the graph...
 3.59: In Exercises 59 to 66, graph each rational function.
 3.60: In Exercises 59 to 66, graph each rational function.
 3.61: In Exercises 59 to 66, graph each rational function.
 3.62: In Exercises 59 to 66, graph each rational function.
 3.63: In Exercises 59 to 66, graph each rational function.
 3.64: In Exercises 59 to 66, graph each rational function.
 3.65: In Exercises 59 to 66, graph each rational function.
 3.66: In Exercises 59 to 66, graph each rational function.
 3.67: Average Cost of Skateboards The cost, in dollars, of producing x sk...
 3.68: Food Temperature The temperature F, in degrees Fahrenheit, of a des...
 3.69: Motor Vehicle Thefts The following table lists the number of motor ...
 3.70: Physiology One of Poiseuilles laws states that the resistance R enc...
Solutions for Chapter 3: College Algebra and Trigonometry 7th Edition
Full solutions for College Algebra and Trigonometry  7th Edition
ISBN: 9781439048603
Solutions for Chapter 3
Get Full SolutionsChapter 3 includes 70 full stepbystep solutions. Since 70 problems in chapter 3 have been answered, more than 5948 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. College Algebra and Trigonometry was written by and is associated to the ISBN: 9781439048603. This textbook survival guide was created for the textbook: College Algebra and Trigonometry, edition: 7.

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!

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

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).

Condition number
cond(A) = c(A) = IIAIlIIAIII = amaxlamin. In Ax = b, the relative change Ilox III Ilx II is less than cond(A) times the relative change Ilob III lib II· Condition numbers measure the sensitivity of the output to change in the input.

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).

Elimination.
A sequence of row operations that reduces A to an upper triangular U or to the reduced form R = rref(A). Then A = LU with multipliers eO in L, or P A = L U with row exchanges in P, or E A = R with an invertible E.

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

Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.

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

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.

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

Multiplicities AM and G M.
The algebraic multiplicity A M of A is the number of times A appears as a root of det(A  AI) = O. The geometric multiplicity GM is the number of independent eigenvectors for A (= dimension of the eigenspace).

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 •

Plane (or hyperplane) in Rn.
Vectors x with aT x = O. Plane is perpendicular to a =1= O.

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

Rayleigh quotient q (x) = X T Ax I x T x for symmetric A: Amin < q (x) < Amax.
Those extremes are reached at the eigenvectors x for Amin(A) and Amax(A).

Similar matrices A and B.
Every B = MI AM has the same eigenvalues as A.

Vector v in Rn.
Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.

Volume of box.
The rows (or the columns) of A generate a box with volume I det(A) I.