Question

1. (l96) fill in the blank.

the rules for distributing in expressions with several variables ______.
a) are different from the rules for distributing in expressions with one variable
b) are the same as the rules for distributing in expressions with one variable
c) depend on the expressions
d) all of the above
e) none of the above

1. (l100) fill in the blank.

“a difference of two squares” means any expression in the form ______.
a) $a^{2}-b^{2}$
b) $a^{2}+b^{2}$
c) $a^{3}-b^{3}$
d) $a^{4}+b^{4}$
e) none of the above

1. (l81) solve by factoring: $x^{2}=-7x$

a) 1, 0
b) -1, 0
c) -1, -7
d) -7, 0
e) none of the above

Explanation:

Step1: Analyze distribution rules

The rules for distributing in expressions with several variables are the same as the rules for distributing in expressions with one - variable. For example, \(a(b + c)=ab+ac\) whether \(b\) and \(c\) are single - variable terms or multi - variable terms. So for the first question, the answer is b.

Step2: Recall the form of difference of two squares

The form of a difference of two squares is \(a^{2}-b^{2}\). So for the second question, the answer is a.

Step3: Solve the equation by factoring

Given \(x^{2}=-7x\), we can rewrite it as \(x^{2}+7x = 0\). Then factor out an \(x\) to get \(x(x + 7)=0\). Using the zero - product property, if \(ab = 0\), then either \(a = 0\) or \(b = 0\). So \(x=0\) or \(x+7 = 0\) (i.e., \(x=-7\)). The solutions are \(x = 0\) and \(x=-7\), so for the third question, the answer is c.

Answer:

1. b) are the same as the rules for distributing in expressions with one variable
2. a) \(a^{2}-b^{2}\)
3. c) \(-1, - 7\)