 45.1: Tiffany said that the polynomial function f(x) 5 x4 1 x2 1 1 cannot...
 45.2: The graph of y 5 x2 2 4x 1 4 is tangent to the xaxis at x 5 2 and ...
 45.3: In 311, each graph shows a polynomial functions from the set of rea...
 45.4: In 311, each graph shows a polynomial functions from the set of rea...
 45.5: In 311, each graph shows a polynomial functions from the set of rea...
 45.6: In 311, each graph shows a polynomial functions from the set of rea...
 45.7: In 311, each graph shows a polynomial functions from the set of rea...
 45.8: In 311, each graph shows a polynomial functions from the set of rea...
 45.9: In 311, each graph shows a polynomial functions from the set of rea...
 45.10: In 311, each graph shows a polynomial functions from the set of rea...
 45.11: In 311, each graph shows a polynomial functions from the set of rea...
 45.12: In 1217, use a graph to find the solution set of each inequality. x...
 45.13: In 1217, use a graph to find the solution set of each inequality. x...
 45.14: In 1217, use a graph to find the solution set of each inequality. x...
 45.15: In 1217, use a graph to find the solution set of each inequality. x...
 45.16: In 1217, use a graph to find the solution set of each inequality. 2...
 45.17: In 1217, use a graph to find the solution set of each inequality. 2...
 45.18: The perimeter of a rectangle is 24 feet. a. If x is the measure of ...
 45.19: The sum of the lengths of the legs of a right triangle is 20 feet. ...
 45.20: A polynomial function of degree three, p(x), intersects the xaxis ...
 45.21: a. Sketch the graph of y 5 x2 . b. Sketch the graph of y 5 x2 1 2 c...
 45.22: a. Sketch the graph of y 5 x2 . b. Sketch the graph of y 5 (x 1 2)2...
 45.23: a. Sketch the graph of y 5 x2 . b. Sketch the graph of y 5 2x2 . c....
 45.24: a. Sketch the graph of y 5 x2 . b. Sketch the graph of y 5 3x2 c. S...
 45.25: For the parabola whose equation is y 5 ax2 1 bx 1 c, the equation o...
Solutions for Chapter 45: Polynomial Functions
Full solutions for Amsco's Algebra 2 and Trigonometry  1st Edition
ISBN: 9781567657029
Solutions for Chapter 45: Polynomial Functions
Get Full SolutionsChapter 45: Polynomial Functions includes 25 full stepbystep solutions. This textbook survival guide was created for the textbook: Amsco's Algebra 2 and Trigonometry, edition: 1. Since 25 problems in chapter 45: Polynomial Functions have been answered, more than 31064 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. Amsco's Algebra 2 and Trigonometry was written by and is associated to the ISBN: 9781567657029.

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.

Back substitution.
Upper triangular systems are solved in reverse order Xn to Xl.

Cholesky factorization
A = CTC = (L.J]))(L.J]))T for positive definite A.

Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).

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.

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.

Diagonalization
A = S1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k SI.

Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.

Identity matrix I (or In).
Diagonal entries = 1, offdiagonal entries = 0.

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

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.

Matrix multiplication AB.
The i, j entry of AB is (row i of A)·(column j of B) = L aikbkj. By columns: Column j of AB = A times column j of B. By rows: row i of A multiplies B. Columns times rows: AB = sum of (column k)(row k). All these equivalent definitions come from the rule that A B times x equals A times B x .

Multiplicities AM and G M.
The algebraic multiplicity A M of A is the number of times A appears as a root of det(A  AI) = O. The geometric multiplicity GM is the number of independent eigenvectors for A (= dimension of the eigenspace).

Outer product uv T
= column times row = rank one matrix.

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.

Pivot.
The diagonal entry (first nonzero) at the time when a row is used in elimination.

Rank r (A)
= number of pivots = dimension of column space = dimension of row space.

Reflection matrix (Householder) Q = I 2uuT.
Unit vector u is reflected to Qu = u. All x intheplanemirroruTx = o have Qx = x. Notice QT = Q1 = Q.

Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.

Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.