- 11-4.1: Find each quotient.
- 11-4.2: Find each quotient.
- 11-4.3: Find each quotient.
- 11-4.4: Find each quotient.
- 11-4.5: Find each quotient.
- 11-4.6: Find each quotient.
- 11-4.7: Express 85 kilometers per hour in meters per second.
- 11-4.8: Express 32 pounds per square foot as pounds per square inch.
- 11-4.9: COOKING Latisha was making candy using a two-quart pan. As she stir...
- 11-4.10: Find each quotient.
- 11-4.11: Find each quotient.
- 11-4.12: Find each quotient.
- 11-4.13: Find each quotient.
- 11-4.14: Find each quotient.
- 11-4.15: Find each quotient.
- 11-4.16: Find each quotient.
- 11-4.17: Find each quotient.
- 11-4.18: Find each quotient.
- 11-4.19: Find each quotient.
- 11-4.20: Find each quotient.
- 11-4.21: Find each quotient.
- 11-4.22: What is the quotient when _2x + 6 x + 5 is divided by _2 x + 5 ?
- 11-4.23: Find the quotient when _ m - 8 m + 7 is divided by m 2 - 7m - 8.
- 11-4.24: Complete.
- 11-4.25: Complete.
- 11-4.26: Complete.
- 11-4.27: Complete.
- 11-4.28: TRIATHLONS Sadie is training for an upcoming triathlon and plans to...
- 11-4.29: VOLUNTEERING Tyrell is passing out orange drink from a 3.5-gallon c...
- 11-4.30: LANDSCAPING For Exercises 30 and 31, use the following information....
- 11-4.31: LANDSCAPING For Exercises 30 and 31, use the following information....
- 11-4.32: TRUCKS For Exercises 32 and 33, use the following information. The ...
- 11-4.33: TRUCKS For Exercises 32 and 33, use the following information. The ...
- 11-4.34: SCULPTURE For Exercises 34 and 35, use the following information. A...
- 11-4.35: SCULPTURE For Exercises 34 and 35, use the following information. A...
- 11-4.36: OPEN ENDED Give an example of a real-world situation that could be ...
- 11-4.37: REASONING Tell whether the following statement is always, sometimes...
- 11-4.38: CHALLENGE Which expression is not equivalent to the reciprocal of x...
- 11-4.39: Writing in Math Use the information about soft drinks and aluminum ...
- 11-4.40: Which expression could be used to represent the width of the rectan...
- 11-4.41: REVIEW What is the surface area of the regular square pyramid? V V ...
- 11-4.42: Find each product.
- 11-4.43: Find each product.
- 11-4.44: Find each product.
- 11-4.45: Simplify each expression.
- 11-4.46: Simplify each expression.
- 11-4.47: Simplify each expression.
- 11-4.48: Simplify each expression.
- 11-4.49: MANUFACTURING Global Sporting Equipment sells tennis racket covers ...
- 11-4.50: PREREQUISITE SKILL Simplify.
- 11-4.51: PREREQUISITE SKILL Simplify.
- 11-4.52: PREREQUISITE SKILL Simplify.
- 11-4.53: PREREQUISITE SKILL Simplify.
- 11-4.54: PREREQUISITE SKILL Simplify.
Solutions for Chapter 11-4: Dividing Rational Expressions
Full solutions for Algebra 1, Student Edition (MERRILL ALGEBRA 1) | 1st Edition
A matrix can be partitioned into matrix blocks, by cuts between rows and/or between columns. Block multiplication ofAB is allowed if the block shapes permit.
Change of basis matrix M.
The old basis vectors v j are combinations L mij Wi of the new basis vectors. The coordinates of CI VI + ... + cnvn = dl wI + ... + dn Wn are related by d = M c. (For n = 2 set VI = mll WI +m21 W2, V2 = m12WI +m22w2.)
Column picture of Ax = b.
The vector b becomes a combination of the columns of A. The system is solvable only when b is in the column space C (A).
Identity matrix I (or In).
Diagonal entries = 1, off-diagonal entries = 0.
Jordan form 1 = M- 1 AM.
If A has s independent eigenvectors, its "generalized" eigenvector matrix M gives 1 = diag(lt, ... , 1s). The block his Akh +Nk where Nk has 1 's on diagonall. Each block has one eigenvalue Ak and one eigenvector.
Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).
Least squares solution X.
The vector x that minimizes the error lie 112 solves AT Ax = ATb. Then e = b - Ax is orthogonal to all columns of A.
Left inverse A+.
If A has full column rank n, then A+ = (AT A)-I AT has A+ A = In.
The pivot row j is multiplied by eij and subtracted from row i to eliminate the i, j entry: eij = (entry to eliminate) / (jth pivot).
Outer product uv T
= column times row = rank one matrix.
Pseudoinverse A+ (Moore-Penrose 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).
Reflection matrix (Householder) Q = I -2uuT.
Unit vector u is reflected to Qu = -u. All x intheplanemirroruTx = o have Qx = x. Notice QT = Q-1 = Q.
Saddle point of I(x}, ... ,xn ).
A point where the first derivatives of I are zero and the second derivative matrix (a2 II aXi ax j = Hessian matrix) is indefinite.
Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.
Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).
Sum V + W of subs paces.
Space of all (v in V) + (w in W). Direct sum: V n W = to}.
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.
Triangle inequality II u + v II < II u II + II v II.
For matrix norms II A + B II < II A II + II B II·
Vector space V.
Set of vectors such that all combinations cv + d w remain within V. Eight required rules are given in Section 3.1 for scalars c, d and vectors v, w.
Stretch and shift the time axis to create Wjk(t) = woo(2j t - k).