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# Solutions for Chapter Chapter 4: Polynomial and Rational Functions

## Full solutions for Precalculus Enhanced with Graphing Utilities | 6th Edition

ISBN: 9780132854351

Solutions for Chapter Chapter 4: Polynomial and Rational Functions

Solutions for Chapter Chapter 4
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##### ISBN: 9780132854351

Precalculus Enhanced with Graphing Utilities was written by and is associated to the ISBN: 9780132854351. Chapter Chapter 4: Polynomial and Rational Functions includes 48 full step-by-step solutions. This expansive textbook survival guide covers the following chapters and their solutions. Since 48 problems in chapter Chapter 4: Polynomial and Rational Functions have been answered, more than 55790 students have viewed full step-by-step solutions from this chapter. This textbook survival guide was created for the textbook: Precalculus Enhanced with Graphing Utilities, edition: 6.

Key Math Terms and definitions covered in this textbook
• 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.

• 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.)

• 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 n-l - An).

• Complex conjugate

z = a - ib for any complex number z = a + ib. Then zz = Iz12.

• Cyclic shift

S. Permutation with S21 = 1, S32 = 1, ... , finally SIn = 1. Its eigenvalues are the nth roots e2lrik/n of 1; eigenvectors are columns of the Fourier matrix F.

• Gauss-Jordan method.

Invert A by row operations on [A I] to reach [I A-I].

• Hessenberg matrix H.

Triangular matrix with one extra nonzero adjacent diagonal.

• Iterative method.

A sequence of steps intended to approach the desired solution.

• Linearly dependent VI, ... , Vn.

A combination other than all Ci = 0 gives L Ci Vi = O.

• Multiplier eij.

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).

• Nullspace N (A)

= All solutions to Ax = O. Dimension n - r = (# columns) - rank.

• Orthogonal subspaces.

Every v in V is orthogonal to every w in W.

• Orthonormal vectors q 1 , ... , q n·

Dot products are q T q j = 0 if i =1= j and q T q i = 1. The matrix Q with these orthonormal columns has Q T Q = I. If m = n then Q T = Q -1 and q 1 ' ... , q n is an orthonormal basis for Rn : every v = L (v T q j )q j •

• Polar decomposition A = Q H.

Orthogonal Q times positive (semi)definite H.

• Projection p = a(aTblaTa) onto the line through a.

P = aaT laTa has rank l.

• Right inverse A+.

If A has full row rank m, then A+ = AT(AAT)-l has AA+ = 1m.

• Schur complement S, D - C A -} B.

Appears in block elimination on [~ g ].

• Solvable system Ax = b.

The right side b is in the column space of A.

• 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.

• Vandermonde matrix V.

V c = b gives coefficients of p(x) = Co + ... + Cn_IXn- 1 with P(Xi) = bi. Vij = (Xi)j-I and det V = product of (Xk - Xi) for k > i.

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