MULTIPLICATION AND DIVISION OF ALGEBRAIC NUMBERS

$- 3 x^{2} y^{2} z^{2}$ , $2 x^{2} y z^{3}$ , and $- 5 x^{4} y z$ .

If $a = - 2$ , $b = 3$ , and $c = - 1$ , find the value of:

$2 a b^{2} - 3 b c^{2} + c$ .

$4 a^{2} - 2 b^{2} - c^{2}$ .

$5 a + 2 b - 4 c^{4}$ .

$2 a^{3} - 3 b + 8 c^{2}$ .

$- a + 3 b - 2 c^{2}$ .

$- a^{3} - 2 b - 10 c$ .

$3 a^{3} - 3 b^{3} - 3 c^{3}$ .

$2 a b^{2} - 3 b c^{2} + 2 a c$ .

$3 a b c + 5 a^{2} b^{2} - 2 a^{2} b$ .

$a b^{2} c^{2} + 2 a b c^{2} + a^{2} b^{2} c^{2}$ .

$2 a^{2} b c + 3 a b c + a^{2} b^{2} c^{2}$ .

$6 a^{2} + 8 a^{2} b^{2} - 5 a^{2} b c$ .

76. Division.

To divide $48$ by $8$ is to find the number of times it is necessary to take $8$ to make $48$ . Here the product and one factor are given, and the other factor is required. We may, therefore, take for the definition of division:

The operation by which when the product and one factor are given, the other factor is found.

With reference to this operation the product is called the dividend, the given factor the divisor, and the required factor the quotient.

77. Law of Signs in Division.

{2} &\text{Since } (+a) × (+b) = +ab,\quad && \therefore +ab ÷ (+a) = +b. \ &\text{Since } (+a) × (-b) = -ab, && \therefore -ab ÷ (+a) = -b. \ &\text{Since } (-a) × (+b) = -ab, && \therefore -ab ÷ (-a) = +b. \ &\text{Since } (-a) × (-b) = +ab, && \therefore +ab ÷ (-a) = -b.

That is, if the dividend and divisor have like signs, the quotient has the plus sign; and if they have unlike signs, the quotient has the minus sign. Hence, in division,

Like signs give $+$ , and unlike signs give $-$ .