 Chapter 8.1: State whether each sentence is true or false. If false, replace the...
 Chapter 8.2: State whether each sentence is true or false. If false, replace the...
 Chapter 8.3: State whether each sentence is true or false. If false, replace the...
 Chapter 8.4: State whether each sentence is true or false. If false, replace the...
 Chapter 8.5: State whether each sentence is true or false. If false, replace the...
 Chapter 8.6: State whether each sentence is true or false. If false, replace the...
 Chapter 8.7: State whether each sentence is true or false. If false, replace the...
 Chapter 8.8: State whether each sentence is true or false. If false, replace the...
 Chapter 8.9: Simplify each expression.
 Chapter 8.10: Simplify each expression.
 Chapter 8.11: Simplify each expression.
 Chapter 8.12: Simplify each expression.
 Chapter 8.13: Simplify each expression.
 Chapter 8.14: Simplify each expression.
 Chapter 8.15: A triangle has an area of 2 x 2 + 4x  16 square meters. If the bas...
 Chapter 8.16: Simplify each expression
 Chapter 8.17: Simplify each expression
 Chapter 8.18: Simplify each expression
 Chapter 8.19: Simplify each expression
 Chapter 8.20: Simplify each expression
 Chapter 8.21: Simplify each expression
 Chapter 8.22: Simplify the equation.
 Chapter 8.23: What would the acid level be after 30 minutes?
 Chapter 8.24: Graph each rational function
 Chapter 8.25: Graph each rational function
 Chapter 8.26: Graph each rational function
 Chapter 8.27: Graph each rational function
 Chapter 8.28: Graph each rational function
 Chapter 8.29: Graph each rational function
 Chapter 8.30: A group makes 45 sandwiches to take on a picnic. The number of sand...
 Chapter 8.31: If y varies directly as x and y = 21 when x = 7, find x when y = 5.
 Chapter 8.32: If y varies inversely as x and y = 9 when x = 2.5, find y when x = ...
 Chapter 8.33: If y varies inversely as x and y = 4 when x = 8, find y when x = ...
 Chapter 8.34: If y varies jointly as x and z and x = 2 and z = 4 when y = 16, fin...
 Chapter 8.35: If y varies jointly as x and z and y = 14 when x = 10 and z = 7, fi...
 Chapter 8.36: Chriss pay varies directly with how many lawns he mows. If his pay ...
 Chapter 8.37: Identify the type of function represented by each graph.
 Chapter 8.38: Identify the type of function represented by each graph.
 Chapter 8.39: Solve each equation or inequality. Check your solutions.
 Chapter 8.40: Solve each equation or inequality. Check your solutions.
 Chapter 8.41: Solve each equation or inequality. Check your solutions.
 Chapter 8.42: Solve each equation or inequality. Check your solutions.
 Chapter 8.43: Solve each equation or inequality. Check your solutions.
 Chapter 8.44: Danielle can put a puzzle together in three hours. Aidan can put th...
Solutions for Chapter Chapter 8: Rational Expressions and Equations
Full solutions for California Algebra 2: Concepts, Skills, and Problem Solving  1st Edition
ISBN: 9780078778568
Solutions for Chapter Chapter 8: Rational Expressions and Equations
Get Full SolutionsChapter Chapter 8: Rational Expressions and Equations includes 44 full stepbystep solutions. This textbook survival guide was created for the textbook: California Algebra 2: Concepts, Skills, and Problem Solving, edition: 1. Since 44 problems in chapter Chapter 8: Rational Expressions and Equations have been answered, more than 42519 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. California Algebra 2: Concepts, Skills, and Problem Solving was written by and is associated to the ISBN: 9780078778568.

Affine transformation
Tv = Av + Vo = linear transformation plus shift.

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

CayleyHamilton Theorem.
peA) = det(A  AI) has peA) = zero matrix.

Characteristic equation det(A  AI) = O.
The n roots are the eigenvalues of A.

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

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].

Determinant IAI = det(A).
Defined by det I = 1, sign reversal for row exchange, and linearity in each row. Then IAI = 0 when A is singular. Also IABI = IAIIBI and

Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then SI AS = A = eigenvalue matrix.

Free variable Xi.
Column i has no pivot in elimination. We can give the n  r free variables any values, then Ax = b determines the r pivot variables (if solvable!).

GaussJordan method.
Invert A by row operations on [A I] to reach [I AI].

Minimal polynomial of A.
The lowest degree polynomial with meA) = zero matrix. This is peA) = det(A  AI) if no eigenvalues are repeated; always meA) divides peA).

Network.
A directed graph that has constants Cl, ... , Cm associated with the edges.

Orthogonal matrix Q.
Square matrix with orthonormal columns, so QT = Ql. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.

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

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

Pseudoinverse A+ (MoorePenrose inverse).
The n by m matrix that "inverts" A from column space back to row space, with N(A+) = N(AT). A+ A and AA+ are the projection matrices onto the row space and column space. Rank(A +) = rank(A).

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.

Rank one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.

Row picture of Ax = b.
Each equation gives a plane in Rn; the planes intersect at x.

Tridiagonal matrix T: tij = 0 if Ii  j I > 1.
T 1 has rank 1 above and below diagonal.