Multiplication and Division of Directed Algebraic Terms

When multiplying and dividing positive and negative numbers remember this: If the signs are the same the answer is positive, if the signs are different the answer is negative.

Even numbers of negative signs (-) give a plus sign (+)

e.g. (–) × (–) × (–) × (–) = +  (There are 4 negative signs which is even)

Odd numbers of negative signs (-) give a negative sign (-)

e.g. (–) × (–) × (–) × (–) × (–) = – (There are 5 negative signs which is odd)

Example 2.2.1:

Simplify the following expressions 

a. \( \scriptsize 4 \: \times \: (+8y) \)
b. \( \scriptsize (-3) \: \times \: (+8y) \)
c. 12ab ÷ (-4a)
d. (-9a) × (-3a) × (-2a) 
e. (-18x³) ÷ (-3x)
f. (-5) × (6a) 

Solution 

a. \( \scriptsize 4 \: \times \: (+8y) \)

= \( \scriptsize 4 \: \times \: 8y \)

= \( \scriptsize 4 \: \times \: 8 \: \times \: y \)

= \( \scriptsize 32y \)

 b. \( \scriptsize (-3) \: \times \: (+8y) \)

= \( \scriptsize -3 \: \times \: 8y \)

= \( \scriptsize -3 \: \times \: 8 \: \times \: y \)

= \( \scriptsize -24y \)

c. 12ab ÷ (-4a) = \( \frac{12ab}{-4a} \\ = \frac{12 \: \times \: a \: \times \: b}{-4\: \times \: a} \\ = \: -\left(\frac{12ab}{4a} \right) \\ = \: – \left(\frac{12\not{a}b}{4\not{a}} \right) \\ = \: – \frac{12b}{4} \\ = \scriptsize -3b \)

d. (-9a) × (-3a) × (-2a) 

= \( \scriptsize \: – (9a\: \times \: 3a \: \times \: 2a) \)

= \( \scriptsize \: – (9\: \times \: 3\: \times \: 2\: \times \:a\: \times \:a \: \times \:a) \)

=\( \scriptsize \: – 54a^3 \)

e. (-18x³) ÷ (-3x)

= \( \frac{-18x^3}{-3x} \\ = \frac{-18 \: \times \: x \: \times \: x\: \times \: \not{x}}{-3 \: \times \: \not{x}}\\ =\frac{-18 \: \times \: x \: \times \: x }{-3 } \\ = \scriptsize 6x^2\)

f. (-5) × (6a)

= -5 × 6 × a

= -30a