 9.4.1: Use the zeroproduct property to solve each equation. a. (x + 4)(x ...
 9.4.2: Graph each equation and then rewrite it in factored form. a. y = x2...
 9.4.3: Name the xintercepts of the parabola described by each quadratic e...
 9.4.4: Write an equation of a quadratic function that corresponds to each ...
 9.4.5: Consider the equation y = (x + 1)(x 3). a. How many xintercepts do...
 9.4.6: Is the expression on the left equivalent to the expression on the r...
 9.4.7: Use a rectangle diagram to factor each expression. a. x2 + 7x + 6 b...
 9.4.8: The sum and product of the roots of a quadratic equation are relate...
 9.4.9: MiniInvestigation In this exercise you will discover whether knowi...
 9.4.10: Write a quadratic equation of a parabola with xintercepts at 3 and...
 9.4.11: APPLICATION The school ecology club wants to fence in an area along...
 9.4.12: MiniInvestigation Consider the equation y = x2 9. a. Graph the equ...
 9.4.13: Kayleigh says that the roots of 0 = x2 + 16 are 4 and 4 because (4)...
 9.4.14: Reduce the rational expressions by dividing out common factors from...
 9.4.15: Multiply and combine like terms. a. (x 21) (x + 2) b. (3x + 1)(x + ...
 9.4.16: Edward is responsible for keeping the stockroom packed with the bes...
Solutions for Chapter 9.4: Factored Form
Full solutions for Discovering Algebra: An Investigative Approach  2nd Edition
ISBN: 9781559537636
Solutions for Chapter 9.4: Factored Form
Get Full SolutionsChapter 9.4: Factored Form includes 16 full stepbystep solutions. Discovering Algebra: An Investigative Approach was written by and is associated to the ISBN: 9781559537636. This expansive textbook survival guide covers the following chapters and their solutions. Since 16 problems in chapter 9.4: Factored Form have been answered, more than 9855 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Discovering Algebra: An Investigative Approach, edition: 2.

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.

Basis for V.
Independent vectors VI, ... , v d whose linear combinations give each vector in V as v = CIVI + ... + CdVd. V has many bases, each basis gives unique c's. A vector space has many bases!

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.

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.

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.

Full row rank r = m.
Independent rows, at least one solution to Ax = b, column space is all of Rm. Full rank means full column rank or full row rank.

Graph G.
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.

Hypercube matrix pl.
Row n + 1 counts corners, edges, faces, ... of a cube in Rn.

Indefinite matrix.
A symmetric matrix with eigenvalues of both signs (+ and  ).

Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = O.

Norm
IIA II. The ".e 2 norm" of A is the maximum ratio II Ax II/l1x II = O"max· Then II Ax II < IIAllllxll and IIABII < IIAIIIIBII and IIA + BII < IIAII + IIBII. Frobenius norm IIAII} = L La~. The.e 1 and.e oo norms are largest column and row sums of laij I.

Normal matrix.
If N NT = NT N, then N has orthonormal (complex) eigenvectors.

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

Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.

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.

Schur complement S, D  C A } B.
Appears in block elimination on [~ g ].

Semidefinite matrix A.
(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.

Singular matrix A.
A square matrix that has no inverse: det(A) = o.

Skewsymmetric matrix K.
The transpose is K, since Kij = Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.

Stiffness matrix
If x gives the movements of the nodes, K x gives the internal forces. K = ATe A where C has spring constants from Hooke's Law and Ax = stretching.