136. A compound expression involving x and y is divisible by xy if the expression vanishes when + y is put for x; and is divisible by x + y if the expression vanishes when

y is put for x.

If n is a positive integer, prove by the Factor Theorem :

1. x + yn is never divisible by x - y.

Put y for x in xn + yn; then xn + yn = yn + yn = 2 уп.
As 2 yn is not zero, xn + yn is not divisible by x
y.

2. χη
y" is divisible by x + y, if n is even.
Put y for x in an
yn, then an yn = (-y)n - yn.
If n is even, (- y)n = y", and (- y)" - yn = yn – yn.
As yn yn = 0, xn
yn is divisible by x + y, if n is even.

3. x + yn is divisible by x + y, if n is odd.

Put y for x in x + yn, then æn + yn = (-y)n + yn.
If n is odd, (-y)" =
yn, and (-y)n + yn =
yn + yn.
As yn + yn = 0, xn + yn is divisible by x + y, if n is odd.

From § 134 and these three cases, we have

1. For all positive integral values of n,

xn

yn = (x - y) (xn-1 + xn-2y + xn-2y2 + ..... + yn-1).

2. For all positive even integral values of n,

xn

yn = (x + y) (xn-1 - xn-2y + xn - 3y2 - ..... - yn-1).

3. For all positive odd integral values of n,

xn + yn = (x + y) (xn-1 - xn-2y + x - 3y2 - ..... + y^-1).

4. xn + yn is never divisible by x-y; and is not divisible by x + y, if n is even.

NOTE. In applying the preceding rules for resolving an expression into factors, if the terms have a common monomial factor, this factor should be removed first.

When an expression can be expressed as the difference of two perfect squares, the method of § 126 should be employed in preference to any other.

81. a2-2ab + b2 + 12 xy - 4x2 — 9 у2.

82. 2x2 - 4 ху + 2y2 + 2 ax - 2 ау.

110.68 - 30 a2b + 36 ab2.
111. 25 а2 - 4x2 + 4x – 10 α.
112. x*y - x2y3 - xy2 + xy2.
113. 9a2 - 4 b2 + 3a + 2b.
114. x3 y3 - (x2 - y2) - (x - y)2.
115. (x - y)2 — 1 — 2 (х — у — 1).
116. α3 3 - 2 a2c + a2 - 4a + 8c — 4.
117. а2 — b2 - c2 + 2 bc + a + b - c.
118. x3z2 - 8 y3z2 - 4 x3n2 + 32 y3n2.
119. 5 ас+ 3 bc + c + 5 ab + 3b2 + 6.
120. 2 ab - 2 bc - ax + cx + 2 b2 - bx.
121. x2-2 abx2 - a2 - a2b2 — 64.

CHAPTER VIII.

COMMON FACTORS AND MULTIPLES.

Highest Common Factor.

137. Common Factors. A common factor of two or more integral numbers in Arithmetic is an integral number that divides each of them without a remainder.

138. A common factor of two or more integral and rational expressions in Algebra is an integral and rational expression that divides each of them without a remainder.

Thus, 5 a is a common factor of 20 a and 25 a.

139. Two numbers in Arithmetic are said to be prime to each other when they have no common factor except 1.

140. Two expressions in Algebra are said to be prime to each other when they have no common factor except 1.

141. The greatest common factor of two or more integral numbers in Arithmetic is the greatest number that will divide each of them without a remainder.

142. The highest common factor of two or more integral and rational expressions in Algebra is an integral and rational expression of highest degree that will divide each of them without a remainder.

Thus, 3 a2 is the highest common factor of 3a2, 6a3, and 12 a4; 5 x2y2 is the highest common factor of 10 x3y2 and 15 x2y2.