- 5.1.1: A polynomial is a single term or the sum of two or more terms conta...
- 5.1.2: It is customary to write the terms of a polynomial in the order of ...
- 5.1.3: A simplified polynomial that has exactly one term is called a/an __...
- 5.1.4: A simplified polynomial that has two terms is called a/an _________...
- 5.1.5: A simplified polynomial that has three terms is called a/an _______...
- 5.1.6: If a 0, the degree of axn is ______________.
- 5.1.7: If a 0, the degree of axn ym is ______________.
- 5.1.8: The degree of a polynomial is the ______________ degree of all the ...
- 5.1.9: True or false: Polynomial functions of degree 2 or higher have grap...
- 5.1.10: True or false: Polynomial functions of degree 2 or higher have brea...
- 5.1.11: The behavior of the graph of a polynomial function to the far left ...
- 5.1.12: The graph of f(x) = x3 ______________ to the left and _____________...
- 5.1.13: The graph of f(x) = -x3 ______________ to the left and ____________...
- 5.1.14: The graph of f(x) = x2 ______________ to the left and _____________...
- 5.1.15: The graph of f(x) = -x2 ______________ to the left and ____________...
- 5.1.16: True or false: Odd-degree polynomial functions have graphs with opp...
- 5.1.17: True or false: Even-degree polynomial functions have graphs with th...
- 5.1.18: Terms of a polynomial that contain the same variables raised to the...
- 5.1.19: -7x3 + 4x3
- 5.1.20: -4x3 y + x3 y
- 5.1.21: x5 + x5
- 5.1.22: 7x5 y2 - (-3x5 y2 )
- 5.1.23: 12xy2 - 12y2
- 5.1.24: In Exercises 2124, identify which graphs are not those of polynomia...
- 5.1.25: f(x) = -x4 + x2
- 5.1.26: f(x) = x3 - 4x2
- 5.1.27: f(x) = x2 - 6x + 9
- 5.1.28: f(x) = -x3 - x2 + 5x - 3
- 5.1.29: (-6x3 + 5x2 - 8x + 9) + (17x3 + 2x2 - 4x - 13)
- 5.1.30: (-7x3 + 6x2 - 11x + 13) + (19x3 - 11x2 + 7x - 17)
- 5.1.31: a 2 5 x4 + 2 3 x3 + 5 8 x2 + 7b + a- 4 5 x4 + 1 3 x3 - 1 4 x2 - 7b
- 5.1.32: a 1 5 x4 + 1 3 x3 + 3 8 x2 + 6b + a- 3 5 x4 + 2 3 x3 - 1 2 x2 - 6b
- 5.1.33: (7x2y - 5xy) + (2x2y - xy)
- 5.1.34: (-4x2y + xy) + (7x2y + 8xy)
- 5.1.35: (5x2y + 9xy + 12) + (-3x2y + 6xy + 3)
- 5.1.36: (8x2y + 12xy + 14) + (-2x2y + 7xy + 4)
- 5.1.37: (9x4y2 - 6x2y2 + 3xy) + (-18x4y2 - 5x2y - xy)
- 5.1.38: (10x4y2 - 3x2y2 + 2xy) + (-16x4y2 - 4x2y - xy)
- 5.1.39: (x2n + 5xn - 8) + (4x2n - 7xn + 2)
- 5.1.40: (6y2n + yn + 5) + (3y2n - 4yn - 15)
- 5.1.41: (17x3 - 5x2 + 4x - 3) - (5x3 - 9x2 - 8x + 11)
- 5.1.42: (18x3 - 2x2 - 7x + 8) - (9x3 - 6x2 - 5x + 7)
- 5.1.43: (13y5 + 9y4 - 5y2 + 3y + 6) - (-9y5 - 7y3 + 8y2 + 11)
- 5.1.44: (12y5 + 7y4 - 3y2 + 6y + 7) - (-10y5 - 8y3 + 3y2 + 14)
- 5.1.45: (x3 + 7xy - 5y2 ) - (6x3 - xy + 4y2 )
- 5.1.46: (x4 - 7xy - 5y3 ) - (6x4 - 3xy + 4y3 )
- 5.1.47: (3x4y2 + 5x3y - 3y) - (2x4y2 - 3x3y - 4y + 6x)
- 5.1.48: (5x4y2 + 6x3y - 7y) - (3x4y2 - 5x3y - 6y + 8x)
- 5.1.49: (7y2n + yn - 4) - (6y2n - yn - 1)
- 5.1.50: (8x2n + xn - 4) - (9x2n - xn - 2)
- 5.1.51: Subtract -5a2b4 - 8ab2 - ab from 3a2b4 - 5ab2 + 7ab.
- 5.1.52: Subtract -7a2b4 - 8ab2 - ab from 13a2b4 - 17ab2 + ab.
- 5.1.53: Subtract -4x3 - x2y + xy2 + 3y3 from x3 + 2x2y - y3 .
- 5.1.54: Subtract -6x3 + x2y - xy2 + 2y3 from x3 + 2xy2 - y3 .
- 5.1.55: Add 6x4 - 5x3 + 2x to the difference between 4x3 + 3x2 - 1 and x4 -...
- 5.1.56: Add 5x4 - 2x3 + 7x to the difference between 2x3 + 5x2 - 3 and -x4 ...
- 5.1.57: Subtract 9x2y2 - 3x2 - 5 from the sum of -6x2y2 - x2 - 1 and 5x2y2 ...
- 5.1.58: Subtract 6x2y3 - 2x2 - 7 from the sum of -5x2y3 + 3x2 - 4 and 4x2y3...
- 5.1.59: (f - g)(x) and (f - g)(-1)
- 5.1.60: (g - h)(x) and (g - h)(-1)
- 5.1.61: (f + g - h)(x) and (f + g - h)(-2)
- 5.1.62: (g + h - f)(x) and (g + h - f)(-2)
- 5.1.63: 2f(x) - 3g(x)
- 5.1.64: -2g(x) - 3h(x)
- 5.1.65: a. Find and interpret f(40). Identify this information as a point o...
- 5.1.66: a. Find and interpret f(10). Identify this information as a point o...
- 5.1.67: Use the Leading Coefficient Test to determine the end behavior to t...
- 5.1.68: Use the Leading Coefficient Test to determine the end behavior to t...
- 5.1.69: The common cold is caused by a rhinovirus. After x days of invasion...
- 5.1.70: The polynomial function f(x) = -0.87x3 + 0.35x2 + 81.62x + 7684.94 ...
- 5.1.71: A herd of 100 elk is introduced to a small island. The number of el...
- 5.1.72: What is a polynomial?
- 5.1.73: Explain how to determine the degree of each term of a polynomial.
- 5.1.74: Explain how to determine the degree of a polynomial.
- 5.1.75: Explain how to determine the leading coefficient of a polynomial.
- 5.1.76: What is a polynomial function?
- 5.1.77: What do we mean when we describe the graph of a polynomial function...
- 5.1.78: What is meant by the end behavior of a polynomial function?
- 5.1.79: Explain how to use the Leading Coefficient Test to determine the en...
- 5.1.80: Why is a polynomial function of degree 3 with a negative leading co...
- 5.1.81: Explain how to add polynomials.
- 5.1.82: Explain how to subtract polynomials.
- 5.1.83: In a favorable habitat and without natural predators, a population ...
- 5.1.84: Write a polynomial function that imitates the end behavior of each ...
- 5.1.85: Write a polynomial function that imitates the end behavior of each ...
- 5.1.86: Write a polynomial function that imitates the end behavior of each ...
- 5.1.87: Write a polynomial function that imitates the end behavior of each ...
- 5.1.88: f(x) = x3 + 13x2 + 10x - 4
- 5.1.89: f(x) = -2x3 + 6x2 + 3x - 1
- 5.1.90: f(x) = -x4 + 8x3 + 4x2 + 2
- 5.1.91: f(x) = -x5 + 5x4 - 6x3 + 2x + 20
- 5.1.92: f(x) = x3 - 6x + 1, g(x) = x3
- 5.1.93: f(x) = -x4 + 2x3 - 6x, g(x) = -x4
- 5.1.94: Many English words have prefixes with meanings similar to those use...
- 5.1.95: I can determine a polynomials leading coefficient by inspecting the...
- 5.1.96: When Im trying to determine end behavior, its the coefficient of th...
- 5.1.97: When I rearrange the terms of a polynomial, its important that I mo...
- 5.1.98: 4x3 + 7x2 - 5x + 2 x is a polynomial containing four terms.
- 5.1.99: If two polynomials of degree 2 are added, the sum must be a polynom...
- 5.1.100: (x2 - 7x) - (x2 - 4x) = -11x for all values of x.
- 5.1.101: All terms of a polynomial are monomials.
- 5.1.102: (x2n - 3xn + 5) + (4x2n - 3xn - 4) - (2x2n - 5xn - 3)
- 5.1.103: (y3n - 7y2n + 3) - (-3y3n - 2y2n - 1) + (6y3n - y2n + 1)
- 5.1.104: From what polynomial must 4x2 + 2x - 3 be subtracted to obtain 5x2 ...
- 5.1.105: Solve: 9(x - 1) = 1 + 3(x - 2). (Section 1.4, Example 3)
- 5.1.106: Graph: 2x - 3y 6 -6. (Section 4.4, Example 1)
- 5.1.107: Write the point-slope form and slope-intercept form of equations of...
- 5.1.108: Multiply: (2x3y2 )(5x4y7 ).
- 5.1.109: Use the distributive property to multiply: 2x4 (8x4 + 3x).
- 5.1.110: Simplify and express the polynomial in standard form: 3x(x2 + 4x + ...
Solutions for Chapter 5.1: Introduction to Polynomials and Polynomial Functions
Full solutions for Intermediate Algebra for College Students | 6th Edition
A = CTC = (L.J]))(L.J]))T for positive definite A.
Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.
Put CI, ... ,Cn in row n and put n - 1 ones just above the main diagonal. Then det(A - AI) = ±(CI + c2A + C3A 2 + .•. + cnA n-l - An).
Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).
cond(A) = c(A) = IIAIlIIA-III = 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.
Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then S-I AS = A = eigenvalue matrix.
A = S-1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k S-I.
Elimination matrix = Elementary matrix Eij.
The identity matrix with an extra -eij in the i, j entry (i #- j). Then Eij A subtracts eij times row j of A from row i.
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.
Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.
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.
Independent vectors VI, .. " vk.
No combination cl VI + ... + qVk = zero vector unless all ci = O. If the v's are the columns of A, the only solution to Ax = 0 is x = o.
Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.
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.
Permutation matrix P.
There are n! orders of 1, ... , n. The n! P 's have the rows of I in those orders. P A puts the rows of A in the same order. P is even or odd (det P = 1 or -1) based on the number of row exchanges to reach I.
Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.
Singular matrix A.
A square matrix that has no inverse: det(A) = o.
Sum V + W of subs paces.
Space of all (v in V) + (w in W). Direct sum: V n W = to}.