 76.1: Find each product. (a + 6) 2
 76.2: Find each product. (2a + 7b) 2
 76.3: Find each product. (3x + 9y) 2
 76.4: Find each product. (4n  3)(4n 3)
 76.5: Find each product. (x 2  6y) 2
 76.6: Find each product. (9  p) 2
 76.7: GEOMETRY The area A of a triangle is half the product of the base b...
 76.8: Find each product.
 76.9: Find each product.
 76.10: Find each product.
 76.11: Find each product.
 76.12: Find each product.
 76.13: Find each product.
 76.14: Find each product.
 76.15: Find each product.
 76.16: Find each product.
 76.17: Find each product.
 76.18: Find each product.
 76.19: Find each product.
 76.20: Find each product.
 76.21: Find each product.
 76.22: Find each product.
 76.23: Find each product.
 76.24: Find each product.
 76.25: Find each product.
 76.26: Find each product.
 76.27: Find each product.
 76.28: Find each product.
 76.29: Find each product.
 76.30: GEOMETRY Write an expression to represent the area of each figure.
 76.31: GEOMETRY Write an expression to represent the area of each figure.
 76.32: GEOMETRY Write an expression to represent the area of each figure.
 76.33: GEOMETRY Write an expression to represent the area of each figure.
 76.34: Find each product.
 76.35: Find each product.
 76.36: Find each product.
 76.37: Find each product.
 76.38: Find each product.
 76.39: Find each product.
 76.40: Simplify.
 76.41: Simplify.
 76.42: GEOMETRY The volume V of a prism equals the area of the base B time...
 76.43: GEOMETRY The volume V of a prism equals the area of the base B time...
 76.44: BASKETBALL The dimensions of a professional basketball court are re...
 76.45: OFFICE SPACE For Exercises 4547, use the following information. LaT...
 76.46: OFFICE SPACE For Exercises 4547, use the following information. LaT...
 76.47: OFFICE SPACE For Exercises 4547, use the following information. LaT...
 76.48: POOL CONSTRUCTION A homeowner is installing a swimming pool in his ...
 76.49: REASONING Compare and contrast the procedure used to multiply a tri...
 76.50: ALGEBRA TILES Draw a diagram to show how you would use algebra tile...
 76.51: CHALLENGE Determine whether the following statement is sometimes, a...
 76.52: OPEN ENDED Write a binomial and a trinomial involving a single vari...
 76.53: Writing in Math Using the information about multiplying binomials o...
 76.54: A rectangles width is represented by x and its length by y. Which e...
 76.55: REVIEW Tanias age is 4 years less than twice her little brother Bil...
 76.56: Simplify.
 76.57: Simplify.
 76.58: GEOMETRY The sum of the degree measures of the angles of a triangle...
 76.59: If f(x) = 2x  5 and g(x) = x 2 + 3x, find each value.
 76.60: If f(x) = 2x  5 and g(x) = x 2 + 3x, find each value.
 76.61: If f(x) = 2x  5 and g(x) = x 2 + 3x, find each value.
 76.62: Solve each equation or formula for the variable specified. (
 76.63: Solve each equation or formula for the variable specified. (
 76.64: Solve each equation or formula for the variable specified. (
 76.65: Solve each equation.
 76.66: Solve each equation.
 76.67: Solve each equation.
 76.68: PREREQUISITE SKILL Simplify.
 76.69: PREREQUISITE SKILL Simplify.
 76.70: PREREQUISITE SKILL Simplify.
 76.71: PREREQUISITE SKILL Simplify.
 76.72: PREREQUISITE SKILL Simplify.
 76.73: PREREQUISITE SKILL Simplify.
Solutions for Chapter 76: Multiplying Polynomials
Full solutions for Algebra 1, Student Edition (MERRILL ALGEBRA 1)  1st Edition
ISBN: 9780078738227
Solutions for Chapter 76: Multiplying Polynomials
Get Full SolutionsThis textbook survival guide was created for the textbook: Algebra 1, Student Edition (MERRILL ALGEBRA 1) , edition: 1. Chapter 76: Multiplying Polynomials includes 73 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. Since 73 problems in chapter 76: Multiplying Polynomials have been answered, more than 37251 students have viewed full stepbystep solutions from this chapter. Algebra 1, Student Edition (MERRILL ALGEBRA 1) was written by and is associated to the ISBN: 9780078738227.

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.

Fibonacci numbers
0,1,1,2,3,5, ... satisfy Fn = Fnl + Fn 2 = (A7 A~)I()q A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].

Free variable Xi.
Column i has no pivot in elimination. We can give the n  r free variables any values, then Ax = b determines the r pivot variables (if solvable!).

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.

Hankel matrix H.
Constant along each antidiagonal; hij depends on i + j.

Incidence matrix of a directed graph.
The m by n edgenode incidence matrix has a row for each edge (node i to node j), with entries 1 and 1 in columns i and j .

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.

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.

Nullspace N (A)
= All solutions to Ax = O. Dimension n  r = (# columns)  rank.

Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.

Rank r (A)
= number of pivots = dimension of column space = dimension of row space.

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

Toeplitz matrix.
Constant down each diagonal = timeinvariant (shiftinvariant) filter.

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

Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and AI 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·

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