 7.1: Write each expression using exponents.
 7.2: Write each expression using exponents.
 7.3: Write each expression using exponents.
 7.4: Write each expression using exponents.
 7.5: Write each expression using exponents.
 7.6: Write each expression using exponents.
 7.7: Write each expression using exponents.
 7.8: Write each expression using exponents.
 7.9: Evaluate each expression.
 7.10: Evaluate each expression.
 7.11: Evaluate each expression.
 7.12: Evaluate each expression.
 7.13: Evaluate each expression.
 7.14: Evaluate each expression.
 7.15: PROBABILITY The probability of correctly guessing the outcome of a ...
 7.16: Find the area or volume of each figure.
 7.17: Find the area or volume of each figure.
 7.18: Find the area or volume of each figure.
 7.19: Find the area or volume of each figure.
 7.20: MULTIPLE CHOICE If three consecutive integers are x, x + 1, and x +...
 7.21: Simplify. Assume that no denominator is equal to zero. ( 3b c _ 2 4...
 7.22: Simplify. Assume that no denominator is equal to zero. x 2 y 0 z 3
 7.23: Simplify. Assume that no denominator is equal to zero. _27 b 2 14 ...
 7.24: Simplify. Assume that no denominator is equal to zero. (3 a 3 b c 2...
 7.25: Simplify. Assume that no denominator is equal to zero. 16 a 3 b 2x...
 7.26: Simplify. Assume that no denominator is equal to zero. (a ) 5 b _ ...
 7.27: Simplify. Assume that no denominator is equal to zero. (4 a 1 ) _ ...
 7.28: Simplify. Assume that no denominator is equal to zero. ( 5xy _ 2 3...
 7.29: Simplify. Assume that no denominator is equal to zero. _12 3 (_ m n...
 7.30: GEOMETRY The area of a triangle is 50 a 2 b square feet. The base o...
 7.31: Find the degree of each polynomial. n  2 p 2
 7.32: Find the degree of each polynomial. 29 n 2 + 17 n 2 t 2
 7.33: Find the degree of each polynomial. 4xy + 9x 3 z 2 + 17rs 3
 7.34: Find the degree of each polynomial. 6x 5 y  2y 4 + 4  8y 2
 7.35: Arrange the terms of each polynomial so that the powers of x are in...
 7.36: Arrange the terms of each polynomial so that the powers of x are in...
 7.37: CONSTRUCTION Ben is building a brick patio with pavers using the dr...
 7.38: Find each sum or difference. (2x 2  5x + 7)  (3x 3 + x 2 + 2)
 7.39: Find each sum or difference. (x 2  6xy + 7y 2 ) + (3x 2 + xy  y 2 )
 7.40: Find each sum or difference. (7z 2 + 4)  (3z 2 + 2z  6)
 7.41: Find each sum or difference. (13m 4  7m  10) + (8m 4  3m + 9)
 7.42: Find each sum or difference. (11m 2 n 2 + 4mn  6) + (5m 2 n 2 + 6m...
 7.43: Find each sum or difference. (5p 2 + 3p + 49)  (2p 2 + 5p + 24)
 7.44: GARDENING Kyle is planting flowers around the perimeter of his rect...
 7.45: Simplify. b(4b  1) + 10b
 7.46: Simplify. x (3x  5) + 7(x 2  2x + 9)
 7.47: Simplify. 8y(11y 2  2y + 13)  9(3y 3  7y + 2)
 7.48: Simplify. 2x(x  y 2 + 5)  5y 2 (3x  2)
 7.49: Solve each equation. m(2m  5) + m = 2m(m  6) + 16
 7.50: Solve each equation. 2(3w + w 2 )  6 = 2w(w  4) + 10
 7.51: SHOPPING Nicole bought x shirts for $15.00 each, y pants for $25.72...
 7.52: Find each product. (r  3)(r + 7)
 7.53: Find each product. (4a  3)(a + 4)
 7.54: Find each product. (5r  7s)(4r + 3s)
 7.55: Find each product. (3x + 0.25)(6x  0.5)
 7.56: Find each product. (2k + 1)(k 2 + 7k  9)
 7.57: Find each product. (4p  3)(3p 2  p + 2)
 7.58: MANUFACTURING A company is designing a box in the shape of a rectan...
 7.59: Find each product. (x  6)(x + 6)
 7.60: Find each product. (4x + 7) 2
 7.61: Find each product. (8x  5) 2
 7.62: Find each product. (5x  3y)(5x + 3y)
 7.63: Find each product. (6a  5b) 2
 7.64: Find each product. (3m + 4n) 2
 7.65: GENETICS Emily and Santos are both able to roll their tongues. Tong...
Solutions for Chapter 7: Polynomials
Full solutions for Algebra 1, Student Edition (MERRILL ALGEBRA 1)  1st Edition
ISBN: 9780078738227
Solutions for Chapter 7: Polynomials
Get Full SolutionsAlgebra 1, Student Edition (MERRILL ALGEBRA 1) was written by and is associated to the ISBN: 9780078738227. This textbook survival guide was created for the textbook: Algebra 1, Student Edition (MERRILL ALGEBRA 1) , edition: 1. Chapter 7: Polynomials includes 65 full stepbystep solutions. Since 65 problems in chapter 7: Polynomials have been answered, more than 34035 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions.

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

Distributive Law
A(B + C) = AB + AC. Add then multiply, or mUltiply then add.

Factorization
A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.

Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).
Use AT for complex A.

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.

Hilbert matrix hilb(n).
Entries HU = 1/(i + j 1) = Jd X i 1 xj1dx. Positive definite but extremely small Amin and large condition number: H is illconditioned.

Inverse matrix AI.
Square matrix with AI A = I and AAl = I. No inverse if det A = 0 and rank(A) < n and Ax = 0 for a nonzero vector x. The inverses of AB and AT are B1 AI and (AI)T. Cofactor formula (Al)ij = Cji! detA.

Nilpotent matrix N.
Some power of N is the zero matrix, N k = o. The only eigenvalue is A = 0 (repeated n times). Examples: triangular matrices with zero diagonal.

Normal matrix.
If N NT = NT N, then N has orthonormal (complex) eigenvectors.

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.

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 •

Rotation matrix
R = [~ CS ] rotates the plane by () and R 1 = RT rotates back by (). Eigenvalues are eiO and eiO , eigenvectors are (1, ±i). c, s = cos (), sin ().

Row space C (AT) = all combinations of rows of A.
Column vectors by convention.

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.

Spanning set.
Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!

Special solutions to As = O.
One free variable is Si = 1, other free variables = o.

Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.

Symmetric matrix A.
The transpose is AT = A, and aU = a ji. AI is also symmetric.

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.

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