117. Sum and Difference of Two Cubes. By performing

RULE 2. The sum of the cubes of two numbers is divisible by the sum of the numbers, and the quotient is the sum of the squares of the numbers minus their product.

RULE 3. The difference of the cubes of two numbers is divisible by the difference of the numbers, and the quotient is the sum of the squares of the numbers plus their product.

118. Sum and Difference of any Two Like Powers. performing the division, we find that

By

When n is a positive integer, it is proved in chap. vii,

1. an+b" is divisible by a + b if n is odd, and by neither

NOTE. It is important to notice in the above examples that the terms of the quotient are all positive when the divisor is a - b, and alternately positive and negative when the divisor is a + b; also, that the quotient is homogeneous, the exponent of a decreasing and of b increasing by 1 for each successive term.

CHAPTER VII.

FACTORS.

119. Rational Expressions.

An expression is rational

if none of its terms contain square or other roots.

By fac

120. Factors of Rational and Integral Expressions. tors of a given integral number in Arithmetic we mean integral numbers that will exactly divide the given number.

Likewise, by factors of a rational and integral expression in Algebra we mean rational and integral expressions that will exactly divide the given expression.

121. Factors of Monomials. The factors of a monomial may be found by inspection. Thus, the factors of 14 ab are 7, 2, a, a, and b.

122. Factors of Polynomials. The form of a polynomial that can be resolved into factors often suggests the process of finding the factors.

123. When the terms have a common monomial factor.

Resolve into factors 2x2+6xy.

Since 2 and x are factors of each term, we have

.. 2x2 + 6xy = 2x (x + 3y).

Hence, the required factors are 2x and x + 3y.

124. When the terms can be grouped so as to show a common compound factor.

1. Resolve into factors ac + ad + bc + bd.

Since one factor is seen in (2) to be c+d, dividing by c + d we obtain the other factor, a + b.

2. Resolve into factors 3x2 + 6 ax + bx +2 ab.

3x2 + 6 ax + bx + 2 ab = (3x2 + 6 ax) + (bx +2 ab)

= 3x (x+2a) + b (x + 2 a)

=(3x+b) (x + 2a).

3. Find the factors of ac + ad· bc bd.

acad-bc-bd= (ac + ad) (bc + bd)

= a(cd)b(c + d)

= (ab) (c + d).

NOTE. Here the last two terms, - bc bd, being put within a parenthesis preceded by the sign have their signs changed to +.

4. Resolve into factors 3 x3 5 x2-6x + 10.

--

3 x3- 5 x2-6x + 10 = (3 x3 — 5 x2) — (6 x — 10)
= x2(3x-5)-2(3x-5)

= (x2-2) (3x-5).

5. Resolve into factors 5 x3 15 ax2 x + 3 a.

5 x3- 15 ax2 - x + 3 a = (5 x3 — 15 ax2) — (x − 3 a)

=5x2(x-3α)-1(x-3a)
=(5x21) (x3a).

6. Resolve into factors 6 y 27 x2y — 10x + 45 x3.

6 y 27x2y-10x + 45 x3