26. a2-4 ab+2ab2-b3 by a2-3 ab+b2.
27. 5 x3- 2x2

3x+4 by 3x2 – 2 x.

28. ax3 x2 + 3 x b by ax3 + x2 + 3 x + b.

30. - 2x"y" + y2" by x2 + 2 x”y" + y2n.
x2n

31. 16 p2+20 pq + 25 q2 by 4p-5q.

39. Multiply a3 — 3 a2b+ 3 ab2 - b3 by a2 - 2 ab+b2, checking the product by homogeneity.

40. Multiply x3 + y2 + z2 + 2 xyz by x + y, checking the product by symmetry.

41. Multiply 3y-7 xy3+5x3y2+4xy by x-y, arranging according to descending powers of a.

50. (a+b) (b + c) − (c + d) (d + a) − (a + c) (b − d).

110. Detached Coefficients can be employed to advantage in the multiplication of polynomials whenever the literal part of the product is easily discovered by inspection. This is evidently the case when two polynomial factors involve but one letter, or when they are homogeneous and involve but two letters.

1. Multiply a2+2ab+b2 by a+b; also by a2 + b2.

.. (a + b) (a2 + 2 ab + b2) = a3 + 3 a2b + 3 ab2 + b3, and (a2 + b2) (a2 + 2 ab + b2) = aa + 2 a3b + 2 a2b2 + 2 ab3 + ba.

1. In (a) it is apparent that the exponents of a decrease by 1, while those of b increase by 1 in each factor. Does this law hold good in the product? When the coefficients are known, can the product be readily written?

2. If any power is lacking in either factor, a zero coefficient must be used when the work is done by the method of detached coefficients, as in (b). It should be observed that x3 + 3x + 5, for example, may be written

+ 0 x2 + 3 x + 5. Why?

15. (a2 + ab + b2) (a3 — a2b + b3) (a — b).

Verify the following identities:

16. (x2+ y2) (m2 + n2) — (mx + ny)2= (my — nx)2. 17. (x− y)(x2+ xy + y2) = (x − y)3 + 3 xy(x − y). 18. (a2+ab+b2)2 — a2b2 = (a2 + ab)2 + (ab + b2)2. 19. a(a — 2 b)3 — b(b − 2 a)3 = (a − b) (a + b)3.

20. (a−b)3+ (b − c)3 + (c − a)3 = 3(a — b) (b − c) (c — a). 21. (x+y+z) (xy + yz + xz) = (x + y) (y + z) (x + z) + xyz.

POWERS

111. As we have already seen, the product of two or more equal factors is called a Power. In this limited sense, a" means the product of m factors, each a; the exponent m therefore is necessarily a positive integer. Hereafter, however, we shall see that the meaning of the word "power" is very much extended.

...

to m factors. The

112. As already stated, am means aaa expression (am)" is taken to mean am × am × am. each am.

Thus, (a2)2 means a2 × a2= a2+2= a1.

...

to n factors,

NOTE. Negative factors should be inclosed in parentheses when there is danger of ambiguity. Thus, (-a)2 means − a x − a = + a2, while

113. In proving the Law of Exponents for multiplication, we established the principle for the product of powers, where the exponents are positive integers. It was shown that

am xa" = am+n ̧

That is, the product of the mth power and the nth power of ɑ equals the (m + n)th power of a.

mn

114. To prove that (aTM)" = aTM", where m and n are positive integers, we proceed as follows:

(a)" = axaxa"... ton factors,
am am

am+m+m... to n terms

=amnamn.

(Law of Exp.)

That is, the nth power of the mth power of a equals the mnth power of a.

...

115. Since (ab)" means (ab) × (ab) × (ab) to m factors, each (ab), it is readily shown that (ab)"a"b", where m is a positive integer. Thus we have

(ab)m=(ab) × (ab) × (ab) ....... to m factors,

...

to m factors) × (bbb ... to m factors), (by Assoc. and Com. Laws)

=am xbm= ambm.

Similarly, the law can be shown to hold for any number of factors. That is,

The mth power of a product equals the product of the mth powers of its factors.

116. Since tax+a+a2, ̄a × ̄a =+a2,

etc., the following laws are evident:

I. All powers of positive numbers are positive.

II. Even powers of negative numbers are positive, but odd powers are negative.

117. Since (a) = a', and (a)2=a, it is readily seen that in raising expressions to the 4th power we may conveniently square the expression and then square that result; and in raising to the 6th power we may first cube, and then square the result.

118. The following products are of such importance and occur so frequently that they demand special attention. By reference to certain type forms we are enabled to write similar products without the labor of performing the actual multiplications. Those most useful in practice are here considered.