 3.4.1: The _______ of _______ states that if f(x) is a polynomial of degre...
 3.4.2: A quadratic factor that cannot be factored as a product of linear f...
 3.4.3: How many linear factors does a polynomial function f of degree n ha...
 3.4.4: Three of the zeros of a fourthdegree polynomial function f are 1, ...
 3.4.5: In Exercises 58, match the function with its exact number of zeros....
 3.4.6: In Exercises 58, match the function with its exact number of zeros....
 3.4.7: In Exercises 58, match the function with its exact number of zeros....
 3.4.8: In Exercises 58, match the function with its exact number of zeros....
 3.4.9: In Exercises 912, confirm that the function has the indicated zeros...
 3.4.10: In Exercises 912, confirm that the function has the indicated zeros...
 3.4.11: In Exercises 912, confirm that the function has the indicated zeros...
 3.4.12: In Exercises 912, confirm that the function has the indicated zeros...
 3.4.13: In Exercises 1316, find all the zeros of the function. Is there a r...
 3.4.14: In Exercises 1316, find all the zeros of the function. Is there a r...
 3.4.15: In Exercises 1316, find all the zeros of the function. Is there a r...
 3.4.16: In Exercises 1316, find all the zeros of the function. Is there a r...
 3.4.17: In Exercises 1736, find all the zeros of the function and write the...
 3.4.18: In Exercises 1736, find all the zeros of the function and write the...
 3.4.19: In Exercises 1736, find all the zeros of the function and write the...
 3.4.20: In Exercises 1736, find all the zeros of the function and write the...
 3.4.21: In Exercises 1736, find all the zeros of the function and write the...
 3.4.22: In Exercises 1736, find all the zeros of the function and write the...
 3.4.23: In Exercises 1736, find all the zeros of the function and write the...
 3.4.24: In Exercises 1736, find all the zeros of the function and write the...
 3.4.25: In Exercises 1736, find all the zeros of the function and write the...
 3.4.26: In Exercises 1736, find all the zeros of the function and write the...
 3.4.27: In Exercises 1736, find all the zeros of the function and write the...
 3.4.28: In Exercises 1736, find all the zeros of the function and write the...
 3.4.29: In Exercises 1736, find all the zeros of the function and write the...
 3.4.30: In Exercises 1736, find all the zeros of the function and write the...
 3.4.31: In Exercises 1736, find all the zeros of the function and write the...
 3.4.32: In Exercises 1736, find all the zeros of the function and write the...
 3.4.33: In Exercises 1736, find all the zeros of the function and write the...
 3.4.34: In Exercises 1736, find all the zeros of the function and write the...
 3.4.35: In Exercises 1736, find all the zeros of the function and write the...
 3.4.36: In Exercises 1736, find all the zeros of the function and write the...
 3.4.37: In Exercises 3744, (a) find all zeros of the function, (b) write th...
 3.4.38: In Exercises 3744, (a) find all zeros of the function, (b) write th...
 3.4.39: In Exercises 3744, (a) find all zeros of the function, (b) write th...
 3.4.40: In Exercises 3744, (a) find all zeros of the function, (b) write th...
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 3.4.42: In Exercises 3744, (a) find all zeros of the function, (b) write th...
 3.4.43: In Exercises 3744, (a) find all zeros of the function, (b) write th...
 3.4.44: In Exercises 3744, (a) find all zeros of the function, (b) write th...
 3.4.45: In Exercises 45 50, find a polynomial function with real coefficien...
 3.4.46: In Exercises 45 50, find a polynomial function with real coefficien...
 3.4.47: In Exercises 45 50, find a polynomial function with real coefficien...
 3.4.48: In Exercises 45 50, find a polynomial function with real coefficien...
 3.4.49: In Exercises 45 50, find a polynomial function with real coefficien...
 3.4.50: In Exercises 45 50, find a polynomial function with real coefficien...
 3.4.51: In Exercises 51 54, a polynomial function f with real coefficients ...
 3.4.52: In Exercises 51 54, a polynomial function f with real coefficients ...
 3.4.53: In Exercises 51 54, a polynomial function f with real coefficients ...
 3.4.54: In Exercises 51 54, a polynomial function f with real coefficients ...
 3.4.55: In Exercises 55 58, write the polynomial (a) as the product of fact...
 3.4.56: In Exercises 55 58, write the polynomial (a) as the product of fact...
 3.4.57: In Exercises 55 58, write the polynomial (a) as the product of fact...
 3.4.58: In Exercises 55 58, write the polynomial (a) as the product of fact...
 3.4.59: In Exercises 59 64, use the given zero to find all the zeros of the...
 3.4.60: In Exercises 59 64, use the given zero to find all the zeros of the...
 3.4.61: In Exercises 59 64, use the given zero to find all the zeros of the...
 3.4.62: In Exercises 59 64, use the given zero to find all the zeros of the...
 3.4.63: In Exercises 59 64, use the given zero to find all the zeros of the...
 3.4.64: In Exercises 59 64, use the given zero to find all the zeros of the...
 3.4.65: In Exercises 65 68, (a) use a graphing utility to find the real zer...
 3.4.66: In Exercises 65 68, (a) use a graphing utility to find the real zer...
 3.4.67: In Exercises 65 68, (a) use a graphing utility to find the real zer...
 3.4.68: In Exercises 65 68, (a) use a graphing utility to find the real zer...
 3.4.69: A football is kicked off the ground with an initial upward velocity...
 3.4.70: The demand equation for a microwave is p = 140 0.001x, where p is t...
 3.4.71: In Exercises 71 and 72, decide whether the statement is true or fal...
 3.4.72: In Exercises 71 and 72, decide whether the statement is true or fal...
 3.4.73: Compile a list of all the various techniques for factoring a polyno...
 3.4.74: Describe a translation of the graph that will result in a function ...
 3.4.75: In Exercises 75 78, describe the graph of the function and identify...
 3.4.76: In Exercises 75 78, describe the graph of the function and identify...
 3.4.77: In Exercises 75 78, describe the graph of the function and identify...
 3.4.78: In Exercises 75 78, describe the graph of the function and identify...
Solutions for Chapter 3.4: Polynomial and Rational Functions
Full solutions for Algebra and Trigonometry: Real Mathematics, Real People  7th Edition
ISBN: 9781305071735
Solutions for Chapter 3.4: Polynomial and Rational Functions
Get Full SolutionsChapter 3.4: Polynomial and Rational Functions includes 78 full stepbystep solutions. Algebra and Trigonometry: Real Mathematics, Real People was written by and is associated to the ISBN: 9781305071735. This textbook survival guide was created for the textbook: Algebra and Trigonometry: Real Mathematics, Real People, edition: 7. Since 78 problems in chapter 3.4: Polynomial and Rational Functions have been answered, more than 59379 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions.

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

Big formula for n by n determinants.
Det(A) is a sum of n! terms. For each term: Multiply one entry from each row and column of A: rows in order 1, ... , nand column order given by a permutation P. Each of the n! P 's has a + or  sign.

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).

Elimination matrix = Elementary matrix Eij.
The identity matrix with an extra eij in the i, j entry (i # j). Then Eij A subtracts eij times row j of A from row i.

Elimination.
A sequence of row operations that reduces A to an upper triangular U or to the reduced form R = rref(A). Then A = LU with multipliers eO in L, or P A = L U with row exchanges in P, or E A = R with an invertible E.

Fourier matrix F.
Entries Fjk = e21Cijk/n give orthogonal columns FT F = nI. Then y = Fe is the (inverse) Discrete Fourier Transform Y j = L cke21Cijk/n.

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.

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.

Linear transformation T.
Each vector V in the input space transforms to T (v) in the output space, and linearity requires T(cv + dw) = c T(v) + d T(w). Examples: Matrix multiplication A v, differentiation and integration in function space.

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.

Positive definite matrix A.
Symmetric matrix with positive eigenvalues and positive pivots. Definition: x T Ax > 0 unless x = O. Then A = LDLT with diag(D» O.

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

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.

Singular Value Decomposition
(SVD) A = U:E VT = (orthogonal) ( diag)( orthogonal) First r columns of U and V are orthonormal bases of C (A) and C (AT), AVi = O'iUi with singular value O'i > O. Last columns are orthonormal bases of nullspaces.

Spectrum of A = the set of eigenvalues {A I, ... , An}.
Spectral radius = max of IAi I.

Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.

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

Vector addition.
v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.

Volume of box.
The rows (or the columns) of A generate a box with volume I det(A) I.