- 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
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.
Eigenvalue A and eigenvector x.
Ax = AX with x#-O so det(A - AI) = o.
0,1,1,2,3,5, ... satisfy Fn = Fn-l + Fn- 2 = (A7 -A~)I()q -A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].
Hilbert matrix hilb(n).
Entries HU = 1/(i + j -1) = Jd X i- 1 xj-1dx. Positive definite but extremely small Amin and large condition number: H is ill-conditioned.
Hypercube matrix pl.
Row n + 1 counts corners, edges, faces, ... of a cube in Rn.
Identity matrix I (or In).
Diagonal entries = 1, off-diagonal entries = 0.
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.
lA-II = 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.
Least squares solution X.
The vector x that minimizes the error lie 112 solves AT Ax = ATb. Then e = b - Ax is orthogonal to all columns of A.
Normal equation AT Ax = ATb.
Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)·(b - Ax) = o.
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.
Rank r (A)
= number of pivots = dimension of column space = dimension of row space.
Row picture of Ax = b.
Each equation gives a plane in Rn; the planes intersect at x.
Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.
Solvable system Ax = b.
The right side b is in the column space of A.
Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.
Vector space V.
Set of vectors such that all combinations cv + d w remain within V. Eight required rules are given in Section 3.1 for scalars c, d and vectors v, w.
Volume of box.
The rows (or the columns) of A generate a box with volume I det(A) I.