 A.4.1: To check division, the expression that is being divided, the divide...
 A.4.2: To divide 2x3  5x + 1 by x + 3 using synthetic division, the first...
 A.4.3: In using synthetic division, the divisor is always a polynomial of ...
 A.4.4: 25 3 2 1 10 14 32 5 7 16 31 means 5x3 + 3x2 + 2x + 1 x + 2 = 5x...
 A.4.5: In 516, use synthetic division to find the quotient and remainder w...
 A.4.6: In 516, use synthetic division to find the quotient and remainder w...
 A.4.7: In 516, use synthetic division to find the quotient and remainder w...
 A.4.8: In 516, use synthetic division to find the quotient and remainder w...
 A.4.9: In 516, use synthetic division to find the quotient and remainder w...
 A.4.10: In 516, use synthetic division to find the quotient and remainder w...
 A.4.11: In 516, use synthetic division to find the quotient and remainder w...
 A.4.12: In 516, use synthetic division to find the quotient and remainder w...
 A.4.13: In 516, use synthetic division to find the quotient and remainder w...
 A.4.14: In 516, use synthetic division to find the quotient and remainder w...
 A.4.15: In 516, use synthetic division to find the quotient and remainder w...
 A.4.16: In 516, use synthetic division to find the quotient and remainder w...
 A.4.17: In 1726, use synthetic division to determine whether x  c is a fac...
 A.4.18: In 1726, use synthetic division to determine whether x  c is a fac...
 A.4.19: In 1726, use synthetic division to determine whether x  c is a fac...
 A.4.20: In 1726, use synthetic division to determine whether x  c is a fac...
 A.4.21: In 1726, use synthetic division to determine whether x  c is a fac...
 A.4.22: In 1726, use synthetic division to determine whether x  c is a fac...
 A.4.23: In 1726, use synthetic division to determine whether x  c is a fac...
 A.4.24: In 1726, use synthetic division to determine whether x  c is a fac...
 A.4.25: In 1726, use synthetic division to determine whether x  c is a fac...
 A.4.26: In 1726, use synthetic division to determine whether x  c is a fac...
 A.4.27: Find the sum of a, b, c, and d if x3  2x2 + 3x + 5 x + 2 = ax2 + b...
 A.4.28: When dividing a polynomial by x  c, do you prefer to use long divi...
Solutions for Chapter A.4: Synthetic Division
Full solutions for Precalculus Enhanced with Graphing Utilities  6th Edition
ISBN: 9780132854351
Solutions for Chapter A.4: Synthetic Division
Get Full SolutionsThis textbook survival guide was created for the textbook: Precalculus Enhanced with Graphing Utilities, edition: 6. Chapter A.4: Synthetic Division includes 28 full stepbystep solutions. Precalculus Enhanced with Graphing Utilities was written by and is associated to the ISBN: 9780132854351. This expansive textbook survival guide covers the following chapters and their solutions. Since 28 problems in chapter A.4: Synthetic Division have been answered, more than 55807 students have viewed full stepbystep solutions from this chapter.

Adjacency matrix of a graph.
Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected). Adjacency matrix of a graph. Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected).

Block matrix.
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.

Companion matrix.
Put CI, ... ,Cn in row n and put n  1 ones just above the main diagonal. Then det(A  AI) = ±(CI + c2A + C3A 2 + .•. + cnA nl  An).

Dot product = Inner product x T y = XI Y 1 + ... + Xn Yn.
Complex dot product is x T Y . Perpendicular vectors have x T y = O. (AB)ij = (row i of A)T(column j of B).

Ellipse (or ellipsoid) x T Ax = 1.
A must be positive definite; the axes of the ellipse are eigenvectors of A, with lengths 1/.JI. (For IIx II = 1 the vectors y = Ax lie on the ellipse IIA1 yll2 = Y T(AAT)1 Y = 1 displayed by eigshow; axis lengths ad

Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.

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.

Fundamental Theorem.
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.

GramSchmidt orthogonalization A = QR.
Independent columns in A, orthonormal columns in Q. Each column q j of Q is a combination of the first j columns of A (and conversely, so R is upper triangular). Convention: diag(R) > o.

lAII = l/lAI and IATI = IAI.
The big formula for det(A) has a sum of n! terms, the cofactor formula uses determinants of size n  1, volume of box = I det( A) I.

Multiplication Ax
= Xl (column 1) + ... + xn(column n) = combination of columns.

Normal equation AT Ax = ATb.
Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)·(b  Ax) = o.

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

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.

Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.

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

Rayleigh quotient q (x) = X T Ax I x T x for symmetric A: Amin < q (x) < Amax.
Those extremes are reached at the eigenvectors x for Amin(A) and Amax(A).

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.

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

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.