 2.2.1: In Exercises 1 and 2, use a graphing utility to graph each function...
 2.2.2: In Exercises 1 and 2, use a graphing utility to graph each function...
 2.2.3: In Exercises 38, sketch the graph of the quadratic function. Identi...
 2.2.4: In Exercises 38, sketch the graph of the quadratic function. Identi...
 2.2.5: In Exercises 38, sketch the graph of the quadratic function. Identi...
 2.2.6: In Exercises 38, sketch the graph of the quadratic function. Identi...
 2.2.7: In Exercises 38, sketch the graph of the quadratic function. Identi...
 2.2.8: In Exercises 38, sketch the graph of the quadratic function. Identi...
 2.2.9: In Exercises 912, write the standard form of the quadratic function...
 2.2.10: In Exercises 912, write the standard form of the quadratic function...
 2.2.11: In Exercises 912, write the standard form of the quadratic function...
 2.2.12: In Exercises 912, write the standard form of the quadratic function...
 2.2.13: Numerical, Graphical, and Analytical Analysis A rectangle is inscri...
 2.2.14: Cost A textile manufacturer has daily production costs of where C i...
 2.2.15: Gardening A gardener has 1500 feet of fencing to enclose three adja...
 2.2.16: Profit An online music company sells songs for $1.75 each. The comp...
 2.2.17: In Exercises 17 and 18, sketch the graph of and each specified tran...
 2.2.18: In Exercises 17 and 18, sketch the graph of and each specified tran...
 2.2.19: Graphical Analysis In Exercises 19 and 20, use a graphing utility t...
 2.2.20: Graphical Analysis In Exercises 19 and 20, use a graphing utility t...
 2.2.21: In Exercises 2124, use the Leading Coefficient Test to describe the...
 2.2.22: In Exercises 2124, use the Leading Coefficient Test to describe the...
 2.2.23: In Exercises 2124, use the Leading Coefficient Test to describe the...
 2.2.24: In Exercises 2124, use the Leading Coefficient Test to describe the...
 2.2.25: In Exercises 2530, (a) find the zeros algebraically, (b) use a grap...
 2.2.26: In Exercises 2530, (a) find the zeros algebraically, (b) use a grap...
 2.2.27: In Exercises 2530, (a) find the zeros algebraically, (b) use a grap...
 2.2.28: In Exercises 2530, (a) find the zeros algebraically, (b) use a grap...
 2.2.29: In Exercises 2530, (a) find the zeros algebraically, (b) use a grap...
 2.2.30: In Exercises 2530, (a) find the zeros algebraically, (b) use a grap...
 2.2.31: In Exercises 3134, find a polynomial function that has the given ze...
 2.2.32: In Exercises 3134, find a polynomial function that has the given ze...
 2.2.33: In Exercises 3134, find a polynomial function that has the given ze...
 2.2.34: In Exercises 3134, find a polynomial function that has the given ze...
 2.2.35: In Exercises 35 and 36, sketch the graph of the function by (a) app...
 2.2.36: In Exercises 35 and 36, sketch the graph of the function by (a) app...
 2.2.37: In Exercises 37 40, (a) use the Intermediate Value Theorem and a gr...
 2.2.38: In Exercises 37 40, (a) use the Intermediate Value Theorem and a gr...
 2.2.39: In Exercises 37 40, (a) use the Intermediate Value Theorem and a gr...
 2.2.40: In Exercises 37 40, (a) use the Intermediate Value Theorem and a gr...
 2.2.41: Graphical Analysis In Exercises 41 44, use a graphing utility to gr...
 2.2.42: Graphical Analysis In Exercises 41 44, use a graphing utility to gr...
 2.2.43: Graphical Analysis In Exercises 41 44, use a graphing utility to gr...
 2.2.44: Graphical Analysis In Exercises 41 44, use a graphing utility to gr...
 2.2.45: In Exercises 4552, use long division to divide.
 2.2.46: In Exercises 4552, use long division to divide.
 2.2.47: In Exercises 4552, use long division to divide.
 2.2.48: In Exercises 4552, use long division to divide.
 2.2.49: In Exercises 4552, use long division to divide.
 2.2.50: In Exercises 4552, use long division to divide.
 2.2.51: In Exercises 4552, use long division to divide.
 2.2.52: In Exercises 4552, use long division to divide.
 2.2.53: In Exercises 53 58, use synthetic division to divide.
 2.2.54: In Exercises 53 58, use synthetic division to divide.
 2.2.55: In Exercises 53 58, use synthetic division to divide.
 2.2.56: In Exercises 53 58, use synthetic division to divide.
 2.2.57: In Exercises 53 58, use synthetic division to divide.
 2.2.58: In Exercises 53 58, use synthetic division to divide.
 2.2.59: In Exercises 59 and 60, use the Remainder Theorem and synthetic div...
 2.2.60: In Exercises 59 and 60, use the Remainder Theorem and synthetic div...
 2.2.61: In Exercises 61 64, (a) verify the given factor(s) of the function ...
 2.2.62: In Exercises 61 64, (a) verify the given factor(s) of the function ...
 2.2.63: In Exercises 61 64, (a) verify the given factor(s) of the function ...
 2.2.64: In Exercises 61 64, (a) verify the given factor(s) of the function ...
 2.2.65: In Exercises 65 and 66, use the Rational Zero Test to list all poss...
 2.2.66: In Exercises 65 and 66, use the Rational Zero Test to list all poss...
 2.2.67: In Exercises 6770, find all the real zeros of the polynomial function.
 2.2.68: In Exercises 6770, find all the real zeros of the polynomial function.
 2.2.69: In Exercises 6770, find all the real zeros of the polynomial function.
 2.2.70: In Exercises 6770, find all the real zeros of the polynomial function.
 2.2.71: In Exercises 71 and 72, use Descartess Rule of Signs to determine t...
 2.2.72: In Exercises 71 and 72, use Descartess Rule of Signs to determine t...
 2.2.73: In Exercises 73 and 74, use synthetic division to verify the upper ...
 2.2.74: In Exercises 73 and 74, use synthetic division to verify the upper ...
 2.2.75: In Exercises 7578, write the complex number in standard form.
 2.2.76: In Exercises 7578, write the complex number in standard form.
 2.2.77: In Exercises 7578, write the complex number in standard form.
 2.2.78: In Exercises 7578, write the complex number in standard form.
 2.2.79: In Exercises 7990, perform the operations and write the result in s...
 2.2.80: In Exercises 7990, perform the operations and write the result in s...
 2.2.81: In Exercises 7990, perform the operations and write the result in s...
 2.2.82: In Exercises 7990, perform the operations and write the result in s...
 2.2.83: In Exercises 7990, perform the operations and write the result in s...
 2.2.84: In Exercises 7990, perform the operations and write the result in s...
 2.2.85: In Exercises 7990, perform the operations and write the result in s...
 2.2.86: In Exercises 7990, perform the operations and write the result in s...
 2.2.87: In Exercises 7990, perform the operations and write the result in s...
 2.2.88: In Exercises 7990, perform the operations and write the result in s...
 2.2.89: In Exercises 7990, perform the operations and write the result in s...
 2.2.90: In Exercises 7990, perform the operations and write the result in s...
 2.2.91: In Exercises 9194, write the quotient in standard form.
 2.2.92: In Exercises 9194, write the quotient in standard form.
 2.2.93: In Exercises 9194, write the quotient in standard form.
 2.2.94: In Exercises 9194, write the quotient in standard form.
 2.2.95: In Exercises 95 and 96, determine the complex number shown in the c...
 2.2.96: In Exercises 95 and 96, determine the complex number shown in the c...
 2.2.97: In Exercises 97102, plot the complex number in the complex plane.
 2.2.98: In Exercises 97102, plot the complex number in the complex plane.
 2.2.99: In Exercises 97102, plot the complex number in the complex plane.
 2.2.100: In Exercises 97102, plot the complex number in the complex plane.
 2.2.101: In Exercises 97102, plot the complex number in the complex plane.
 2.2.102: In Exercises 97102, plot the complex number in the complex plane.
 2.2.103: In Exercises 103 and 104, find all the zeros of the function.
 2.2.104: In Exercises 103 and 104, find all the zeros of the function.
 2.2.105: In Exercises 105110, find all the zeros of the function and write t...
 2.2.106: In Exercises 105110, find all the zeros of the function and write t...
 2.2.107: In Exercises 105110, find all the zeros of the function and write t...
 2.2.108: In Exercises 105110, find all the zeros of the function and write t...
 2.2.109: In Exercises 105110, find all the zeros of the function and write t...
 2.2.110: In Exercises 105110, find all the zeros of the function and write t...
 2.2.111: In Exercises 111 116, (a) find all the zeros of the function, (b) w...
 2.2.112: In Exercises 111 116, (a) find all the zeros of the function, (b) w...
 2.2.113: In Exercises 111 116, (a) find all the zeros of the function, (b) w...
 2.2.114: In Exercises 111 116, (a) find all the zeros of the function, (b) w...
 2.2.115: In Exercises 111 116, (a) find all the zeros of the function, (b) w...
 2.2.116: In Exercises 111 116, (a) find all the zeros of the function, (b) w...
 2.2.117: In Exercises 117120, find a polynomial function with real coefficie...
 2.2.118: In Exercises 117120, find a polynomial function with real coefficie...
 2.2.119: In Exercises 117120, find a polynomial function with real coefficie...
 2.2.120: In Exercises 117120, find a polynomial function with real coefficie...
 2.2.121: In Exercises 121 and 122, write the polynomial (a) as the product o...
 2.2.122: In Exercises 121 and 122, write the polynomial (a) as the product o...
 2.2.123: In Exercises 123 and 124, Use the given zero to find all the zeros ...
 2.2.124: In Exercises 123 and 124, Use the given zero to find all the zeros ...
 2.2.125: In Exercises 125136, (a) find the domain of the function, (b) decid...
 2.2.126: In Exercises 125136, (a) find the domain of the function, (b) decid...
 2.2.127: In Exercises 125136, (a) find the domain of the function, (b) decid...
 2.2.128: In Exercises 125136, (a) find the domain of the function, (b) decid...
 2.2.129: In Exercises 125136, (a) find the domain of the function, (b) decid...
 2.2.130: In Exercises 125136, (a) find the domain of the function, (b) decid...
 2.2.131: In Exercises 125136, (a) find the domain of the function, (b) decid...
 2.2.132: In Exercises 125136, (a) find the domain of the function, (b) decid...
 2.2.133: In Exercises 125136, (a) find the domain of the function, (b) decid...
 2.2.134: In Exercises 125136, (a) find the domain of the function, (b) decid...
 2.2.135: In Exercises 125136, (a) find the domain of the function, (b) decid...
 2.2.136: In Exercises 125136, (a) find the domain of the function, (b) decid...
 2.2.137: Seizure of Illegal Drugs The cost C (in millions of dollars) for th...
 2.2.138: Wildlife A biology class performs an experiment comparing the quant...
 2.2.139: In Exercises 139144, find all of the vertical, horizontal, and slan...
 2.2.140: In Exercises 139144, find all of the vertical, horizontal, and slan...
 2.2.141: In Exercises 139144, find all of the vertical, horizontal, and slan...
 2.2.142: In Exercises 139144, find all of the vertical, horizontal, and slan...
 2.2.143: In Exercises 139144, find all of the vertical, horizontal, and slan...
 2.2.144: In Exercises 139144, find all of the vertical, horizontal, and slan...
 2.2.145: In Exercises 145156, sketch the graph of the rational function by h...
 2.2.146: In Exercises 145156, sketch the graph of the rational function by h...
 2.2.147: In Exercises 145156, sketch the graph of the rational function by h...
 2.2.148: In Exercises 145156, sketch the graph of the rational function by h...
 2.2.149: In Exercises 145156, sketch the graph of the rational function by h...
 2.2.150: In Exercises 145156, sketch the graph of the rational function by h...
 2.2.151: In Exercises 145156, sketch the graph of the rational function by h...
 2.2.152: In Exercises 145156, sketch the graph of the rational function by h...
 2.2.153: In Exercises 145156, sketch the graph of the rational function by h...
 2.2.154: In Exercises 145156, sketch the graph of the rational function by h...
 2.2.155: In Exercises 145156, sketch the graph of the rational function by h...
 2.2.156: In Exercises 145156, sketch the graph of the rational function by h...
 2.2.157: Wildlife The Parks and Wildlife Commission introduces 80,000 fish i...
 2.2.158: Page Design A page that is x inches wide and y inches high contains...
 2.2.159: In Exercises 159162, determine whether the scatter plot could best ...
 2.2.160: In Exercises 159162, determine whether the scatter plot could best ...
 2.2.161: In Exercises 159162, determine whether the scatter plot could best ...
 2.2.162: In Exercises 159162, determine whether the scatter plot could best ...
 2.2.163: Investment The table shows the prices per fine ounce of gold (in do...
 2.2.164: Broccoli The table shows the per capita consumptions (in pounds) of...
 2.2.165: True or False? In Exercises 165 167, determine whether the statemen...
 2.2.166: True or False? In Exercises 165 167, determine whether the statemen...
 2.2.167: True or False? In Exercises 165 167, determine whether the statemen...
 2.2.168: Think About It What does it mean for a divisor to divide evenly int...
 2.2.169: Writing Write a paragraph discussing whether every rational functio...
 2.2.170: Error Analysis Describe the error.
 2.2.171: Error Analysis Describe the error.
 2.2.172: Write each of the powers of as 1, or (a) (b) (c) (d) i 67 i 50 i 25...
 2.2.1: Identify the vertex and intercepts of the graph ofy x 2 4x 3.
 2.2.2: Write an equation of the parabola shown at the right.
 2.2.3: Find all the real zeros of Determine the multiplicity of each zero
 2.2.4: Sketch the graph of the functionfx x3 7x 6. fx
 2.2.5: Divide using long division:3x 3 4x 1 x 2 1. fx
 2.2.6: Divide using synthetic division:2x 4 5x 2 3 x 2. 3x 3
 2.2.7: Use synthetic division to evaluate f2 2 for fx 3x4 6x 5x 1. 2x 4 5
 2.2.8: In Exercises 8 and 9, list all the possible rational zeros of the f...
 2.2.9: In Exercises 8 and 9, list all the possible rational zeros of the f...
 2.2.10: Find all the zeros of the function and write the polynomial as the ...
 2.2.11: In Exercises 11 14, perform the operations and write the result in ...
 2.2.12: In Exercises 11 14, perform the operations and write the result in ...
 2.2.13: In Exercises 11 14, perform the operations and write the result in ...
 2.2.14: In Exercises 11 14, perform the operations and write the result in ...
 2.2.15: In Exercises 1517, write the quotient in standard form.
 2.2.16: In Exercises 1517, write the quotient in standard form.
 2.2.17: In Exercises 1517, write the quotient in standard form.
 2.2.18: Plot the complex number in the complex plane.
 2.2.19: In Exercises 1921, sketch the graph of the rational function. As sk...
 2.2.20: In Exercises 1921, sketch the graph of the rational function. As sk...
 2.2.21: In Exercises 1921, sketch the graph of the rational function. As sk...
 2.2.22: The table shows the amounts A (in billions of dollars) budgeted for...
Solutions for Chapter 2: Polynomial and Rational Functions
Full solutions for Precalculus With Limits A Graphing Approach  5th Edition
ISBN: 9780618851522
Solutions for Chapter 2: Polynomial and Rational Functions
Get Full SolutionsThis textbook survival guide was created for the textbook: Precalculus With Limits A Graphing Approach, edition: 5. Since 194 problems in chapter 2: Polynomial and Rational Functions have been answered, more than 33110 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 2: Polynomial and Rational Functions includes 194 full stepbystep solutions. Precalculus With Limits A Graphing Approach was written by and is associated to the ISBN: 9780618851522.

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!

Change of basis matrix M.
The old basis vectors v j are combinations L mij Wi of the new basis vectors. The coordinates of CI VI + ... + cnvn = dl wI + ... + dn Wn are related by d = M c. (For n = 2 set VI = mll WI +m21 W2, V2 = m12WI +m22w2.)

Conjugate Gradient Method.
A sequence of steps (end of Chapter 9) to solve positive definite Ax = b by minimizing !x T Ax  x Tb over growing Krylov subspaces.

Determinant IAI = det(A).
Defined by det I = 1, sign reversal for row exchange, and linearity in each row. Then IAI = 0 when A is singular. Also IABI = IAIIBI and

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

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.

Indefinite matrix.
A symmetric matrix with eigenvalues of both signs (+ and  ).

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.

Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , AjIb. Numerical methods approximate A I b by x j with residual b  Ax j in this subspace. A good basis for K j requires only multiplication by A at each step.

Network.
A directed graph that has constants Cl, ... , Cm associated with the edges.

Normal matrix.
If N NT = NT N, then N has orthonormal (complex) eigenvectors.

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

Pivot.
The diagonal entry (first nonzero) at the time when a row is used in elimination.

Positive definite matrix A.
Symmetric matrix with positive eigenvalues and positive pivots. Definition: x T Ax > 0 unless x = O. Then A = LDLT with diag(D» O.

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.

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 ().

Semidefinite matrix A.
(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.

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

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

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