594. If we divide 23-y3 by x-y, the quotient is x2 + xy + y2. a3 b3 (ab) (a2 + ab + b2).

=

The difference of the cubes of two quantities may be divided by the difference of the quantities. The quotient consists of three terms, namely: the square of the first, plus the product of the first and second, plus the square of the second. (a3 + b3) ÷ (a + b) = a2 — ab + b2.

Give a general statement for the quotient obtained by dividing the sum of the cubes of two quantities by the sum of the quantities.

596. The following supplementary exercises are applications of the preceding cases.

Sometimes it is possible to

apply more than one case to an exercise.

1. Factor 2 a1x - 2 a2x3 - 24 x5.

First divide by 2x.

but

2 ax2 ax3-24x52x(α-a2x2 - 12 x1),

=

a1- a2x2 - 12 x = (a2 + 3x2) (a2 − 4 x2); hence 2 a1x-2 a2x23 – 24 x3 = 2 x (a2 + 3 x2) (a2 — 4 x2),

but

a2 - 4x2= (a + 2x) (α-2x);

hence 2ax-2 a2x3- 24 a5 = 2x(a2 + 3x2) (a + 2x) (α-2x).

2. Factor 26— yo.

Since x6 = (3)2, and y = (y3)2,

x6 — y3 = (x3 + y3) (23 — y3).

Factoring the second member of the above equation,

x6y= (x + y) (x2 — xy + y2) (x − y) (x2 + xy + y2).

When in doubt prove your work by either multiplication or division.

The value of a fraction is not changed when both numerator and denominator are either multiplied or divided by the same quantity.

6. Change +2 to a fraction whose denominator is x+3.

x+2=x+2

+2. Multiply the numerator and the denominator by x+3.

What may be done with the numerators when the denominator is