- P.3.1: For the polynomial the degree is ________, the leading coefficient ...
- P.3.2: A polynomial in in standard form is written with ________ powers of
- P.3.3: A polynomial with one term is called a ________, while a polynomial...
- P.3.4: To add or subtract polynomials, add or subtract the ________ ______...
- P.3.5: The letters in FOIL stand for the following. F ________ O ________ ...
- P.3.6: u v u v u2 v2 x
- P.3.7: u v2 u2 2uv v2
- P.3.8: u v2 u2 2uv v2
- P.3.9: A polynomial of degree 0
- P.3.10: A trinomial of degree 5
- P.3.11: A binomial with leading coefficient
- P.3.12: A monomial of positive degree
- P.3.13: A trinomial with leading coefficient
- P.3.14: A third-degree polynomial with leading coefficient 1
- P.3.15: A third-degree polynomial with leading coefficient
- P.3.16: A fifth-degree polynomial with leading coefficient 6
- P.3.17: A fourth-degree binomial with a negative leading coefficient
- P.3.18: A third-degree binomial with an even leading coefficient
- P.3.19: 14x 2
- P.3.20: 2x
- P.3.21: x
- P.3.22: 7x
- P.3.23: 3 x
- P.3.24: y 25y
- P.3.25: 3
- P.3.26: 8 t 2 3
- P.3.27: 6x
- P.3.28: 3 2x
- P.3.29: 4x3y
- P.3.30: x5y 2x2y2 xy 4 4
- P.3.31: 2x 3x 3 8
- P.3.32: 5x4 2x2 x2 2x
- P.3.33: 3x 4 x
- P.3.34: x2 2x 3 2
- P.3.35: y
- P.3.36: y
- P.3.37: 6x 5 8x 15
- P.3.38: 2x 2 1 x 2 2x 1
- P.3.39: t 3 1 6t 3 5t 2
- P.3.40: 5x 2 1 3x 2 5 t
- P.3.41: 15x 2 6 8.3x 3 14.7x 2 17 5x
- P.3.42: 15.6w4 14w 17.4 16.9w4 9.2w 13 15
- P.3.43: 5z 3z 10z 8
- P.3.44: y 3 1 y 2 1 3y 7 5
- P.3.45: 3x
- P.3.46: y2 4y
- P.3.47: 5z 3z 1 3
- P.3.48: 3x 5x 2
- P.3.49: 1 x 3 4x
- P.3.50: 4x 3 x 3
- P.3.51: 1.5t 2 5 3t
- P.3.52: 2 3.5y 2y3
- P.3.53: 2x 0.1x 17 y
- P.3.54: 6y 5 3 8 2
- P.3.55: Add
- P.3.56: Add
- P.3.57: Subtrac
- P.3.58: Subtract
- P.3.59: Multiply
- P.3.60: Multiply
- P.3.61: Multiply
- P.3.62: Multiply
- P.3.63: x 3 x 4
- P.3.64: x 5 x 10
- P.3.65: 3x 5 2x 1
- P.3.66: 7x 2 4x 3
- P.3.67: x 10 x 10
- P.3.68: 2x 3 2x 3
- P.3.69: x 2y x 2y
- P.3.70: 4a 5b 4a 5b
- P.3.71: 2x 32
- P.3.72: 5 8x2
- P.3.73: x 13
- P.3.74: x 23
- P.3.75: 2x y3
- P.3.76: 3x 2y3
- P.3.77: 4x3 32
- P.3.78: 8x 32
- P.3.79: x 2 x 1 x 2 x 1
- P.3.80: x 2 3x 2 x 2 3x 2
- P.3.81: x2 x 5 3x2 4x 1
- P.3.82: 2x2 x 4 x2 3x 2
- P.3.83: m 3 n m 3n
- P.3.84: x 3y z x 3yz
- P.3.85: x 3 y 2
- P.3.86: x 1y 2
- P.3.87: 2r 2 5 2r 2 5
- P.3.88: 3a3 4b2 3a3 4b
- P.3.89: 1 4 x 5 2
- P.3.90: 3 5t 4 2
- P.3.91: 1 5x 3 1 5x 3
- P.3.92: 3x 1 6 3x 1 6
- P.3.93: 2.4x 32
- P.3.94: 1.8y 52
- P.3.95: 1.5x 4 1.5x 4
- P.3.96: 2.5y 3 2.5y 3
- P.3.97: 5x x 13x x 1
- P.3.98: 2x 1 x 3 3 x 3
- P.3.99: u 2 u 2 u2 4
- P.3.100: x y x y x 2 y 2
- P.3.101: x y x y x
- P.3.102: 5 x 5 x
- P.3.103: x 5 2
- P.3.104: x 3
- P.3.105: COST, REVENUE, AND PROFIT An electronics manufacturer can produce a...
- P.3.106: COST, REVENUE, AND PROFIT An artisan can produce and sell hats per ...
- P.3.107: COMPOUND INTEREST After 2 years, an investment of $500 compounded a...
- P.3.108: COMPOUND INTEREST After 3 years, an investment of $1200 compounded ...
- P.3.109: VOLUME OF A BOX A take-out fast-food restaurant is constructing an ...
- P.3.110: VOLUME OF A BOX An overnight shipping company is designing a closed...
- P.3.111: GEOMETRY Find the area of the shaded region in each figure. Write y...
- P.3.112: GEOMETRY Find the area of the shaded region in each figure. Write y...
- P.3.113: GEOMETRY In Exercises 113 and 114, find a polynomial that represent...
- P.3.114: GEOMETRY In Exercises 113 and 114, find a polynomial that represent...
- P.3.115: ENGINEERING A uniformly distributed load is placed on a one-inch-wi...
- P.3.116: STOPPING DISTANCE The stopping distance of an automobile is the dis...
- P.3.117: In Exercises 117 and 118, use the area model to write two different...
- P.3.118: In Exercises 117 and 118, use the area model to write two different...
- P.3.119: The product of two binomials is always a seconddegree polynomial.
- P.3.120: The sum of two binomials is always a binomial.
- P.3.121: Find the degree of the product of two polynomials of
- P.3.122: Find the degree of the sum of two polynomials of degrees and if
- P.3.123: WRITING A students homework paper included the following. Write a p...
- P.3.124: CAPSTONE
- P.3.125: THINK ABOUT IT Must the sum of two seconddegree polynomials be a se...
- P.3.126: THINK ABOUT IT When the polynomial is subtracted from an unknown po...
- P.3.127: LOGICAL REASONING Verify that is not equal to by letting and and ev...
Solutions for Chapter P.3: Polynomials and Special Products
Full solutions for Algebra and Trigonometry | 8th Edition
Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.
Upper triangular systems are solved in reverse order Xn to Xl.
Big formula for n by n determinants.
Det(A) is a sum of n! terms. For each term: Multiply one entry from each row and column of A: rows in order 1, ... , nand column order given by a permutation P. Each of the n! P 's has a + or - sign.
Circulant matrix C.
Constant diagonals wrap around as in cyclic shift S. Every circulant is Col + CIS + ... + Cn_lSn - l . Cx = convolution c * x. Eigenvectors in F.
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).
z = a - ib for any complex number z = a + ib. Then zz = Iz12.
Eigenvalue A and eigenvector x.
Ax = AX with x#-O so det(A - AI) = o.
Free columns of A.
Columns without pivots; these are combinations of earlier columns.
Full row rank r = m.
Independent rows, at least one solution to Ax = b, column space is all of Rm. Full rank means full column rank or full row rank.
Inverse matrix A-I.
Square matrix with A-I A = I and AA-l = 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 B-1 A-I and (A-I)T. Cofactor formula (A-l)ij = Cji! detA.
Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = O.
A directed graph that has constants Cl, ... , Cm associated with the edges.
Nilpotent matrix N.
Some power of N is the zero matrix, N k = o. The only eigenvalue is A = 0 (repeated n times). Examples: triangular matrices with zero diagonal.
Nullspace N (A)
= All solutions to Ax = O. Dimension n - r = (# columns) - rank.
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.
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.
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.
Semidefinite matrix A.
(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.
Constant down each diagonal = time-invariant (shift-invariant) filter.
Stretch and shift the time axis to create Wjk(t) = woo(2j t - k).