 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: Odddegree polynomial functions have graphs with opp...
 5.1.17: True or false: Evendegree 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 pointslope form and slopeintercept 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
ISBN: 9780321758934
Solutions for Chapter 5.1: Introduction to Polynomials and Polynomial Functions
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Intermediate Algebra for College Students, edition: 6. Since 110 problems in chapter 5.1: Introduction to Polynomials and Polynomial Functions have been answered, more than 29606 students have viewed full stepbystep solutions from this chapter. Chapter 5.1: Introduction to Polynomials and Polynomial Functions includes 110 full stepbystep solutions. Intermediate Algebra for College Students was written by and is associated to the ISBN: 9780321758934.

Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.

Augmented matrix [A b].
Ax = b is solvable when b is in the column space of A; then [A b] has the same rank as A. Elimination on [A b] keeps equations correct.

Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.

Companion matrix.
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 nl  An).

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.

Cyclic shift
S. Permutation with S21 = 1, S32 = 1, ... , finally SIn = 1. Its eigenvalues are the nth roots e2lrik/n of 1; eigenvectors are columns of the Fourier matrix F.

Dimension of vector space
dim(V) = number of vectors in any basis for V.

Eigenvalue A and eigenvector x.
Ax = AX with x#O so det(A  AI) = o.

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].

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.

Iterative method.
A sequence of steps intended to approach the desired solution.

Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).

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.

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).

Right inverse A+.
If A has full row rank m, then A+ = AT(AAT)l has AA+ = 1m.

Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).

Subspace S of V.
Any vector space inside V, including V and Z = {zero vector only}.

Symmetric matrix A.
The transpose is AT = A, and aU = a ji. AI is also symmetric.

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