15. x2-4ax+4a2-b2+2by—y2.
16. y2+2by+b2-a2-6ax-9x2.
17. x2-2x+1-a2-4ab-4b2.
18. 9a2-6a+1-x2-8dx-16d2.
19. x2-a2+ y2— b2-2xy+2ab.
20. a2+b2-2ab-c2-d2-2cd.
21. 4x2-12ax-c2-k2-2ck+9a2.

22. a2+6bx-9b2x2-10ab-1+25b2.
23. a4-25x6 +8α2x2-9+30x3+ 16x4.

WHEN AN EXPRESSION IS THE SUM OR DIFFERENCE

OF TWO CUBES.

89. If we divide as+b8 by a+b the quotient is a2-ab+b2; and if we divide a3 – b3 by a−b the quotient is a2+ab+b2. We have therefore the following identities :

a3+b3=(a+b)(a2-ab+b2);

a3-b3= (a−b) (a2+ab+b2).

These results are very important, and enable us to resolve into factors any expression which can be written as the sum or the difference of two cubes.

NOTE. The middle term 6xy is the product of 2x and 3y.

Example 2.

64a3 +1=(4a)3 +(1)3

= (4a+1) (16a2 — 4a+1).

We may usually omit the intermediate step and write down the factors at once.

Examples.

(1) 343a6-27x3 = (7a2 – 3x) (49a1 +21a2x+9x2).

(2) 8x9+729= (2x3 + 9) (4x6 – 18x3 +81).

EXAMPLES XI. k.

Resolve into factors:

1. x3-ys. 2. 23+y3. 5. 8.23-y3. 6. 23+873. 9. a3b3-c3. 10. 8x3+27y3. 13. 125+a3. 14. 216-a3.

3. 3-1. 4. 1+a3. 7. 2723+1. 8. 1-8y3. 11. 1-343x3. 12. 64+y3. 15. a3b3+512. 16. 1000y3-1.

40. x3у3 — 512.

38. p3q3-27.x3. 39. 23-64.

90. Before concluding this chapter we shall draw attention to a few miscellaneous cases of resolution into factors.

Example 1. Resolve into factors 16a-8164.

16a4-81b4=(4a2+9b2) (4a2 – 9b2)

= (4a2 +9b2) (2a+3b) (2a − 3b).

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

NOTE. When an expression can be arranged either as the difference of two squares, or as the difference of two cubes, each of the methods explained in Arts. 86, 89 will be applicable. It will, however, be found simplest to first use the rule for resolving into factors the difference of two squares.

In all cases where an expression to be resolved contains a simple factor common to each of its terms, this should be first taken outside a bracket as explained in Art. 79.

Example 3. Resolve into factors 28x4y+64x3y – 60x2y.

28x4y+64x3y - 60x2y=4x2y (7x2 + 16x - 15)

Example 4. Resolve into factors x3p2 — 8y3μ2 — 4x3q2 +32y3q2.
The expression=p2 (x3- 8y3) - 4q2 (x3 — 8y3)

Example 5. Resolve into factors 4x2 - 25y2+2x+5y.

4x2 - 25y2+2x+5y= (2x+5y) (2x − 5y)+2x+5y

=

= (2x+5y) (2x − 5y + 1).

24. x4+4x2y22+4y4z4. 25. a3b3+512. 26. 2x2+17x+35.

CHAPTER XII.

HIGHEST COMMON FACTOR.

SIMPLE EXPRESSIONS.

91. DEFINITION. The highest common factor of two or more algebraical expressions is the expression of highest dimensions [Art. 10] which divides each of them without remainder.

The abbreviation H. C. F. is sometimes used instead of the words highest common factor.

92. In the case of simple expressions the highest common factor can be written down by inspection.

Example 1. The highest common factor of a1, a3, a2, a® is a2.

Example 2. The highest common factor of a3b4, ab3c2, a2bc is ab1; for a is the highest power of a that will divide a3, a, a2; b4 is the highest power of b that will divide b1, b5, b7; and c is not a common factor.

93. If the expressions have numerical coefficients, find by Arithmetic their greatest common measure, and prefix it as a coefficient to the algebraical highest common factor.

Example. The highest common factor of 21a1x3y, 35a2x1y, 28a3xy1 is 7a2xy; for it consists of the product of

(1) the numerical greatest common measure of the coefficients; (2) the highest power of each letter which divides every one of the given expressions.

94. We have explained how to write by inspection the highest common factor of two or more simple expressions. An analogous method will enable us readily to find the highest common factor of compound expressions which are given as the product of factors, or which can be easily resolved into factors.

Example 1. Find the highest common factor of

4cx3 and 2cx3 + 4c2x2.

It will be easy to pick out the common factors if the expressions are arranged as follows:

4cx3-4cx3,

2cx3+4c2x2=2cx2 (x+2c);

therefore the H. C. F. is 2cx2.

Example 2. Find the highest common factor of 3a2+9ab, a3-9ab2, a3+6a2b+9ab2.

Resolving each expression into its factors, we have
3a2+9ab=3a (a+3b),

a3-9ab2=
?=a (a+3b) (a − 3b),

a3+6a2b+9ab2=a (a+3b) (a+3b);

therefore the H. C. F. is a (a+3b).

95. When there are two or more expressions containing different powers of the same compound factor, the student should be careful to notice that the highest common factor must contain the highest power of the compound factor which is common to all the given expressions.

Example 1. The highest common factor of

x (a - x)2, a (a-x)3, and 2ax (a− x)5 is (a-x)2.