- 6-4.1: 5x6 - 8x2
- 6-4.2: 2b + 4b3 - 3b5 - 7
- 6-4.3: p(x) = -x3 + x2 - x
- 6-4.4: p(x) = x4 - 3x3 + 2x2 - 5x + 1
- 6-4.5: BIOLOGY The intensity of light emitted by a firefly can be determin...
- 6-4.6: p(a3)
- 6-4.7: 5[q(2a)]
- 6-4.8: 3p(a) - q(a + 1)
- 6-4.9: x f
- 6-4.10: O x f
- 6-4.11: x f(
- 6-4.12: 7 - x
- 6-4.13: (a + 1)(a2 - 4)
- 6-4.14: a2 + 2ab + b2
- 6-4.15: c2 + c - _1 c
- 6-4.16: 6x4 + 3x2 + 4x - 8
- 6-4.17: 7 + 3x2 - 5x3 + 6x2 - 2x
- 6-4.18: p(x) = 2 - x
- 6-4.19: p(x) = x2 - 3x + 8
- 6-4.20: p(x) = 2x3 - x2 + 5x - 7
- 6-4.21: p(x) = x5 - x2
- 6-4.22: r(3a)
- 6-4.23: 4p(a)
- 6-4.24: p(a2)
- 6-4.25: p(2a3)
- 6-4.26: r(x + 1)
- 6-4.27: p(x2 + 3)
- 6-4.28: For each graph, a. describe the end behavior, b. determine whether ...
- 6-4.29: For each graph, a. describe the end behavior, b. determine whether ...
- 6-4.30: For each graph, a. describe the end behavior, b. determine whether ...
- 6-4.31: For each graph, a. describe the end behavior, b. determine whether ...
- 6-4.32: For each graph, a. describe the end behavior, b. determine whether ...
- 6-4.33: For each graph, a. describe the end behavior, b. determine whether ...
- 6-4.34: ENERGY The power generated by a windmill is a function of the speed...
- 6-4.35: PHYSICS For a moving object with mass m in kilograms, the kinetic e...
- 6-4.36: p(x) = x4 - 7x3 + 8x - 6
- 6-4.37: p(x) = 7x2 - 9x + 10
- 6-4.38: p(x) = _1 2 x4 - 2x2 + 4
- 6-4.39: p(x) = _1 8 x3 - _1 4 x2 - _1 2 x + 5
- 6-4.40: 2[p(x + 4)]
- 6-4.41: r(x + 1) - r(x2)
- 6-4.42: 3[p(x2 - 1)] + 4p(x)
- 6-4.43: Is the graph an odd-degree or even-degree function?
- 6-4.44: Discuss the end behavior
- 6-4.45: Do you think attendance at Broadway plays will increase or decrease...
- 6-4.46: The number of regions formed by connecting n points of a circle can...
- 6-4.47: Find the number of regions formed by connecting 5 points of a circl...
- 6-4.48: How many points would you have to connect to form 99 regions?
- 6-4.49: REASONING Explain why a constant polynomial such as f(x) = 4 has de...
- 6-4.50: OPEN ENDED Sketch the graph of an odd-degree polynomial function wi...
- 6-4.51: REASONING Determine whether the following statement is always, some...
- 6-4.52: Find the value of a.
- 6-4.53: For what value(s) of x will f(x) = 0?
- 6-4.54: Simplify and rewrite the function as a cubic function.
- 6-4.55: Sketch the graph of the function.
- 6-4.56: Writing in Math Use the information on page 331 to explain where po...
- 6-4.57: ACT/SAT The figure at the right shows the graph of a polynomial fun...
- 6-4.58: REVIEW Which polynomial represents (4x2 + 5x - 3)(2x - 7)? F 8x3 - ...
- 6-4.59: (t3 - 3t + 2) (t + 2)
- 6-4.60: (y2 + 4y + 3)(y + 1)-1
- 6-4.61: x3 - 3x __2 + 2x - 6 x - 3
- 6-4.62: 3x4 + x3 - 8x __2 + 10x - 3 3x - 2
- 6-4.63: BUSINESS Ms. Schifflet is writing a computer program to find the sa...
- 6-4.64: x2 - 8x - 2 = 0
- 6-4.65: x2 + _1 3 x - _35 36 = 0
- 6-4.66: Write an absolute value inequality for each graph.
- 6-4.67: Write an absolute value inequality for each graph.
- 6-4.68: Write an absolute value inequality for each graph.
- 6-4.69: Write an absolute value inequality for each graph.
- 6-4.70: If 3x = 4y and 4y = 15z, then 3x = 15z.
- 6-4.71: 5y(4a - 6b) = 20ay - 30by
- 6-4.72: 2 + (3 + x) = (2 + 3) + x
- 6-4.73: y = x2 + 4
- 6-4.74: y = -x2 + 6x - 5
- 6-4.75: y = _1 2 x2 + 2x - 6
Solutions for Chapter 6-4: Polynomial Functions
Full solutions for Algebra 2, Student Edition (MERRILL ALGEBRA 2) | 1st Edition
Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).
z = a - ib for any complex number z = a + ib. Then zz = Iz12.
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.
Cross product u xv in R3:
Vector perpendicular to u and v, length Ilullllvlll sin el = area of parallelogram, u x v = "determinant" of [i j k; UI U2 U3; VI V2 V3].
Dimension of vector space
dim(V) = number of vectors in any basis for V.
Echelon matrix U.
The first nonzero entry (the pivot) in each row comes in a later column than the pivot in the previous row. All zero rows come last.
Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.
The nullspace N (A) and row space C (AT) are orthogonal complements in Rn(perpendicular from Ax = 0 with dimensions rand n - r). Applied to AT, the column space C(A) is the orthogonal complement of N(AT) in Rm.
Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).
Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.
Linear transformation T.
Each vector V in the input space transforms to T (v) in the output space, and linearity requires T(cv + dw) = c T(v) + d T(w). Examples: Matrix multiplication A v, differentiation and integration in function space.
= Xl (column 1) + ... + xn(column n) = combination of columns.
Nullspace matrix N.
The columns of N are the n - r special solutions to As = O.
Outer product uv T
= column times row = rank one matrix.
Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.
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.
Symmetric matrix A.
The transpose is AT = A, and aU = a ji. A-I is also symmetric.
Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and A-I are BT AT and (AT)-I.
Tridiagonal matrix T: tij = 0 if Ii - j I > 1.
T- 1 has rank 1 above and below diagonal.
Volume of box.
The rows (or the columns) of A generate a box with volume I det(A) I.