 5.3.1: The process of writing a polynomial containing the sum of monomials...
 5.3.2: An expression with the greatest coefficient and of the highest degr...
 5.3.3: True or false: The factorization of 12x4 + 21x2 is 3x2 # 4x2 + 3x2 ...
 5.3.4: We factor 2x3 + 10x2  6x by factoring out ______________.
 5.3.5: True or false: The factorization of x3  4x2 + 5x  20 is x2 (x  4...
 5.3.6: x3 + 7x2
 5.3.7: 12x4  8x2
 5.3.8: 20x4  8x2
 5.3.9: 32x4 + 2x3 + 8x2
 5.3.10: 9x4 + 18x3 + 6x2
 5.3.11: 4x2y3 + 6xy
 5.3.12: 6x3y2 + 9xy
 5.3.13: 30x2y3  10xy2
 5.3.14: 27x2y3  18xy2
 5.3.15: 12xy  6xz + 4xw
 5.3.16: 14xy  10xz + 8xw
 5.3.17: 15x3y6  9x4y4 + 12x2y5
 5.3.18: 15x4y6  3x3y5 + 12x4y4
 5.3.19: 25x3y6z2  15x4y4z4 + 25x2y5z3
 5.3.20: 49x4y3z5  70x3y5z4 + 35x4y4z3
 5.3.21: 15x2n  25xn
 5.3.22: 12x3n  9x2n
 5.3.23: 4x + 12
 5.3.24: 5x + 20
 5.3.25: 8x  48
 5.3.26: 7x  63
 5.3.27: 2x2 + 6x  14
 5.3.28: 2x2 + 8x  12
 5.3.29: 5y2 + 40x
 5.3.30: 9y2 + 45x
 5.3.31: 4x3 + 32x2  20x
 5.3.32: 5x3 + 50x2  10x
 5.3.33: x2  7x + 5
 5.3.34: x2  8x + 8
 5.3.35: 4(x + 3) + a(x + 3)
 5.3.36: 5(x + 4) + a(x + 4)
 5.3.37: x(y  6)  7(y  6)
 5.3.38: x(y  9)  5(y  9)
 5.3.39: 3x(x + y)  (x + y)
 5.3.40: 7x(x + y)  (x + y)
 5.3.41: 4x2 (3x  1) + 3x  1
 5.3.42: 6x2 (5x  1) + 5x  1
 5.3.43: (x + 2)(x + 3) + (x  1)(x + 3)
 5.3.44: (x + 4)(x + 5) + (x  1)(x + 5)
 5.3.45: x2 + 3x + 5x + 15
 5.3.46: x2 + 2x + 4x + 8
 5.3.47: x2 + 7x  4x  28
 5.3.48: x2 + 3x  5x  15
 5.3.49: x3  3x2 + 4x  12
 5.3.50: x3  2x2 + 5x  10
 5.3.51: xy  6x + 2y  12
 5.3.52: xy  5x + 9y  45
 5.3.53: xy + x  7y  7
 5.3.54: xy + x  5y  5
 5.3.55: 10x2  12xy + 35xy  42y2
 5.3.56: 3x2  6xy + 5xy  10y2
 5.3.57: 4x3  x2  12x + 3
 5.3.58: 3x3  2x2  6x + 4
 5.3.59: x2  ax  bx + ab
 5.3.60: x2 + ax  bx  ab
 5.3.61: x3  12  3x2 + 4x
 5.3.62: 2x3  10 + 4x2  5x
 5.3.63: ay  by + bx  ax
 5.3.64: cx  dx + dy  cy
 5.3.65: ay2 + 2by2  3ax  6bx
 5.3.66: 3a2x + 6a2y  2bx  4by
 5.3.67: xnyn + 3xn + yn + 3
 5.3.68: xnyn  xn + 2yn  2
 5.3.69: ab  c  ac + b
 5.3.70: ab  3c  ac + 3b
 5.3.71: x3  5 + 4x3y  20y
 5.3.72: x3  2 + 3x3y  6y
 5.3.73: 2y7 (3x  1) 5  7y6 (3x  1) 4
 5.3.74: 3y9 (3x  2) 7  5y8 (3x  2) 6
 5.3.75: ax2 + 5ax  2a + bx2 + 5bx  2b
 5.3.76: ax2 + 3ax  11a + bx2 + 3bx  11b
 5.3.77: ax + ay + az  bx  by  bz + cx + cy + cz
 5.3.78: ax2 + ay2  az2 + bx2 + by2  bz2 + cx2 + cy2  cz2
 5.3.79: A ball is thrown straight upward. The function f(t) = 16t 2 + 40t ...
 5.3.80: An explosion causes debris to rise vertically. The function f(t) = ...
 5.3.81: Your computer store is having an incredible sale. The price on one ...
 5.3.82: Your local electronics store is having an endoftheyear sale. The...
 5.3.83: After 2 years, the balance, A, in an account with principal P and i...
 5.3.84: After 3 years, the balance, A, in an account with principal P and i...
 5.3.85: The area of the skating rink with semicircular ends shown is A = pr...
 5.3.86: The amount of sheet metal needed to manufacture a cylindrical tin c...
 5.3.87: What is factoring?
 5.3.88: If a polynomial has a greatest common factor other than 1, explain ...
 5.3.89: Using an example, explain how to factor out the greatest common fac...
 5.3.90: Suppose that a polynomial contains four terms and can be factored b...
 5.3.91: Use two different groupings to factor ac  ad + bd  bc in two ways...
 5.3.92: Write a sentence that uses the word factor as a noun. Then write a ...
 5.3.93: x2  4x = x(x  4)
 5.3.94: x2  2x + 5x  10 = (x  2)(x  5)
 5.3.95: x2 + 2x + x + 2 = x(x + 2) + 1
 5.3.96: x3  3x2 + 4x  12 = (x2 + 4)(x  3)
 5.3.97: After Ive factored a polynomial, my answer cannot always be checked...
 5.3.98: The word greatest in greatest common factor is helpful because it t...
 5.3.99: Although 20x3 appears as a term in both 20x3 + 8x2 and 20x3 + 10x, ...
 5.3.100: You grouped the polynomials terms using different groupings than I ...
 5.3.101: Because the GCF of 9x3 + 6x2 + 3x is 3x, it is not necessary to wri...
 5.3.102: Some polynomials with four terms, such as x3 + x2 + 4x  4, cannot ...
 5.3.103: The polynomial 28x3  7x2 + 36x  9 can be factored by grouping ter...
 5.3.104: x2  2 is a factor of 2  50x  x2 + 25x3 .
 5.3.105: x4n + x2n + x3n
 5.3.106: 3x3mym  6x2my2m
 5.3.107: 8y2n+4 + 16y2n+3  12y2n
 5.3.108: The polynomial has three terms and can be factored using a greatest...
 5.3.109: The polynomial has four terms and can be factored by grouping.
 5.3.110: Solve by Cramers rule: e 3x  2y = 8 2x  5y = 10. (Section 3.5, Ex...
 5.3.111: Determine whether each relation is a function. a. {(0, 5), (3, 5),...
 5.3.112: The length of a rectangle is 2 feet greater than twice its width. I...
 5.3.113: (x + 3)(x + ? ) = x2 + 7x + 12
 5.3.114: (x  ? )(x  12) = x2  14x + 24
 5.3.115: (x + 3y)(x  ? y) = x2  4xy  21y2
Solutions for Chapter 5.3: Greatest Common Factors and Factoring by Grouping
Full solutions for Intermediate Algebra for College Students  6th Edition
ISBN: 9780321758934
Solutions for Chapter 5.3: Greatest Common Factors and Factoring by Grouping
Get Full SolutionsIntermediate Algebra for College Students was written by and is associated to the ISBN: 9780321758934. This textbook survival guide was created for the textbook: Intermediate Algebra for College Students, edition: 6. Chapter 5.3: Greatest Common Factors and Factoring by Grouping includes 115 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. Since 115 problems in chapter 5.3: Greatest Common Factors and Factoring by Grouping have been answered, more than 32358 students have viewed full stepbystep solutions from this chapter.

Adjacency matrix of a graph.
Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected). Adjacency matrix of a graph. Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected).

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!

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].

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!).

Graph G.
Set of n nodes connected pairwise by m edges. A complete graph has all n(n  1)/2 edges between nodes. A tree has only n  1 edges and no closed loops.

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.

Length II x II.
Square root of x T x (Pythagoras in n dimensions).

Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.

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

Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.

Pseudoinverse A+ (MoorePenrose inverse).
The n by m matrix that "inverts" A from column space back to row space, with N(A+) = N(AT). A+ A and AA+ are the projection matrices onto the row space and column space. Rank(A +) = rank(A).

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.

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).

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.

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

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

Spanning set.
Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!

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

Unitary matrix UH = U T = UI.
Orthonormal columns (complex analog of Q).

Wavelets Wjk(t).
Stretch and shift the time axis to create Wjk(t) = woo(2j t  k).