 5.7.1: An equation that can be written in the standard form ax2 + bx + c =...
 5.7.2: The zeroproduct principle states that if AB = 0, then ______________.
 5.7.3: The solutions of ax2 + bx + c = 0 correspond to the ______________ ...
 5.7.4: The equation 5x2 = 20x can be written in standard form by _________...
 5.7.5: The equation x2 + 4 = 8x  12 can be written in standard form by __...
 5.7.6: The result of setting two polynomials equal to each other is called...
 5.7.7: A triangle with one angle measuring 90 is called a/an _____________...
 5.7.8: The Pythagorean Theorem states that in any ______________ triangle,...
 5.7.9: 3x2 = 2  5x
 5.7.10: 5x2 = 2 + 3x
 5.7.11: x2 = 8x
 5.7.12: x2 = 4x
 5.7.13: 3x2 = 5x
 5.7.14: 2x2 = 5x
 5.7.15: x2 + 4x + 4 = 0
 5.7.16: x2 + 6x + 9 = 0
 5.7.17: x2 = 14x  49
 5.7.18: x2 = 12x  36
 5.7.19: 9x2 = 30x  25
 5.7.20: 4x2 = 12x  9
 5.7.21: x2  25 = 0
 5.7.22: x2  49 = 0
 5.7.23: 9x2 = 100
 5.7.24: 4x2 = 25
 5.7.25: x(x  3) = 18
 5.7.26: x(x  4) = 21
 5.7.27: (x  3)(x + 8) = 30
 5.7.28: (x  1)(x + 4) = 14
 5.7.29: x(x + 8) = 16(x  1)
 5.7.30: x(x + 9) = 4(2x + 5)
 5.7.31: (x + 1) 2  5(x + 2) = 3x + 7
 5.7.32: (x + 1) 2 = 2(x + 5)
 5.7.33: x(8x + 1) = 3x2  2x + 2
 5.7.34: 2x(x + 3) = 5x  15
 5.7.35: x2 18 + x 2 + 1 = 0
 5.7.36: x2 4  5x 2 + 6 = 0
 5.7.37: x3 + 4x2  25x  100 = 0
 5.7.38: x3  2x2  x + 2 = 0
 5.7.39: x3  x2 = 25x  25
 5.7.40: x3 + 2x2 = 16x + 32
 5.7.41: 3x4  48x2 = 0
 5.7.42: 5x4  20x2 = 0
 5.7.43: x4  4x3 + 4x2 = 0
 5.7.44: x4  6x3 + 9x2 = 0
 5.7.45: 2x3 + 16x2 + 30x = 0
 5.7.46: 3x3  9x2  30x = 0
 5.7.47: y = x2  6x + 8
 5.7.48: y = x2  2x  8
 5.7.49: y = x2 + 6x + 8
 5.7.50: y = x2 + 2x  8
 5.7.51: x(x + 1) 3  42(x + 1) 2 = 0
 5.7.52: x(x  2) 3  35(x  2) 2 = 0
 5.7.53: 4x[x(3x  2)  8](25x2  40x + 16) = 0
 5.7.54: 7x[x(2x  5)  12](9x2 + 30x + 25) = 0
 5.7.55: f(x) = x2  4x  27 and f(c) = 5.
 5.7.56: f(x) = 5x2  11x + 6 and f(c) = 4.
 5.7.57: f(x) = 2x3 + x2  8x + 2 and f(c) = 6.
 5.7.58: f(x) = x3 + 4x2  x + 6 and f(c) = 10.
 5.7.59: The product of the number decreased by 1 and increased by 4 is 24.
 5.7.60: The product of the number decreased by 6 and increased by 2 is 20.
 5.7.61: If 5 is subtracted from 3 times the number, the result is the squar...
 5.7.62: If the square of the number is subtracted from 61, the result is th...
 5.7.63: f(x) = 3 x2 + 4x  45
 5.7.64: f(x) = 7 x2  3x  28
 5.7.65: How long will it take the gymnast to reach the ground? Use this inf...
 5.7.66: When will the gymnast be 8 feet above the ground? Identify the solu...
 5.7.67: In a roundrobin chess tournament, 21 games were played. How many p...
 5.7.68: In a roundrobin chess tournament, 36 games were played. How many p...
 5.7.69: Identify your solution to Exercise 67 as a point on the graph.
 5.7.70: Identify your solution to Exercise 68 as a point on the graph.
 5.7.71: The length of a rectangular sign is 3 feet longer than the width. I...
 5.7.72: A rectangular parking lot has a length that is 3 yards greater than...
 5.7.73: Each side of a square is lengthened by 3 inches. The area of this n...
 5.7.74: Each side of a square is lengthened by 2 inches. The area of this n...
 5.7.75: A pool measuring 10 meters by 20 meters is surrounded by a path of ...
 5.7.76: A vacant rectangular lot is being turned into a community vegetable...
 5.7.77: As part of a landscaping project, you put in a flower bed measuring...
 5.7.78: As part of a landscaping project, you put in a flower bed measuring...
 5.7.79: A machine produces open boxes using square sheets of metal. The fig...
 5.7.80: A machine produces open boxes using square sheets of metal. The mac...
 5.7.81: The rectangular floor of a closet is divided into two right triangl...
 5.7.82: A piece of wire measuring 20 feet is attached to a telephone pole a...
 5.7.83: A tree is supported by a wire anchored in the ground 15 feet from i...
 5.7.84: A tree is supported by a wire anchored in the ground 5 feet from it...
 5.7.85: What is a quadratic equation?
 5.7.86: What is the zeroproduct principle?
 5.7.87: Explain how to solve x2  x = 6.
 5.7.88: Describe the relationship between the solutions of a quadratic equa...
 5.7.89: What is a polynomial equation? When is it in standard form?
 5.7.90: What is the degree of a polynomial equation? What are polynomial eq...
 5.7.91: Explain how to solve x3 + x2 = x + 1.
 5.7.92: If something is thrown straight up, or possibly dropped, describe a...
 5.7.93: A toy rocket is launched vertically upward. Using a quadratic equat...
 5.7.94: Describe a situation in which a landscape designer might use polyno...
 5.7.95: In your own words, state the Pythagorean Theorem.
 5.7.96: Use the graph of y = x2 + 3x  4 to solve x2 + 3x  4 = 0.
 5.7.97: Use the graph of y = x3 + 3x2  x  3 to solve x3 + 3x2  x  3 = 0.
 5.7.98: Use the graph of y = 2x3  3x2  11x + 6 to solve 2x3  3x2  11x +...
 5.7.99: Use the graph of y = x4 + 4x3  4x2 to solve x4 + 4x3  4x2 = 0.
 5.7.100: Use the TABLE feature of a graphing utility to verify the solution ...
 5.7.101: Im working with a quadratic function that describes the length of t...
 5.7.102: I set the quadratic equation 2x2  5x = 12 equal to zero and obtain...
 5.7.103: Because some trinomials are prime, some quadratic equations cannot ...
 5.7.104: Im looking at a graph with one x@intercept, so it must be the graph...
 5.7.105: Quadratic equations solved by factoring always have two different s...
 5.7.106: If 4x(x2 + 49) = 0, then 4x = 0 or x2 + 49 = 0 x = 0 or x = 7 or x ...
 5.7.107: If 4 is a solution of 7y2 + (2k  5)y  20 = 0, then k must equal 14.
 5.7.108: Some quadratic equations have more than two solutions.
 5.7.109: Write a quadratic equation in standard form whose solutions are 3 ...
 5.7.110: Solve: x2 + 2x  36 = 12.
 5.7.111: Solve: 3x  2 = 8. (Section 4.3, Example 1)
 5.7.112: Simplify: 3(5  7) 2 + 216 + 12 , (3). (Section 1.2, Example 7)
 5.7.113: You invested $3000 in two accounts paying 5% and 8% annual interest...
 5.7.114: If f(x) = 120x 100  x , find f(20).
 5.7.115: Find the domain of f(x) = 4 x  2 .
 5.7.116: Factor the numerator and the denominator. Then simplify by dividing...
Solutions for Chapter 5.7: Polynomial Equations and Their Applications
Full solutions for Intermediate Algebra for College Students  6th Edition
ISBN: 9780321758934
Solutions for Chapter 5.7: Polynomial Equations and Their Applications
Get Full SolutionsThis textbook survival guide was created for the textbook: Intermediate Algebra for College Students, edition: 6. Since 116 problems in chapter 5.7: Polynomial Equations and Their Applications have been answered, more than 9517 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. Intermediate Algebra for College Students was written by and is associated to the ISBN: 9780321758934. Chapter 5.7: Polynomial Equations and Their Applications includes 116 full stepbystep solutions.

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

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.

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.

Factorization
A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.

Fibonacci numbers
0,1,1,2,3,5, ... satisfy Fn = Fnl + Fn 2 = (A7 A~)I()q A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].

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

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.

Inverse matrix AI.
Square matrix with AI A = I and AAl = I. No inverse if det A = 0 and rank(A) < n and Ax = 0 for a nonzero vector x. The inverses of AB and AT are B1 AI and (AI)T. Cofactor formula (Al)ij = Cji! detA.

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.

lAII = l/lAI and IATI = IAI.
The big formula for det(A) has a sum of n! terms, the cofactor formula uses determinants of size n  1, volume of box = I det( A) I.

Left nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.

Lucas numbers
Ln = 2,J, 3, 4, ... satisfy Ln = L n l +Ln 2 = A1 +A~, with AI, A2 = (1 ± /5)/2 from the Fibonacci matrix U~]' Compare Lo = 2 with Fo = O.

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

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

Orthogonal matrix Q.
Square matrix with orthonormal columns, so QT = Ql. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.

Random matrix rand(n) or randn(n).
MATLAB creates a matrix with random entries, uniformly distributed on [0 1] for rand and standard normal distribution for randn.

Reflection matrix (Householder) Q = I 2uuT.
Unit vector u is reflected to Qu = u. All x intheplanemirroruTx = o have Qx = x. Notice QT = Q1 = Q.

Schur complement S, D  C A } B.
Appears in block elimination on [~ g ].

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

Stiffness matrix
If x gives the movements of the nodes, K x gives the internal forces. K = ATe A where C has spring constants from Hooke's Law and Ax = stretching.
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