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CHAPTER XII

INVOLUTION

215. Involution is the operation of raising a quantity to a positive integral power.

To find (3 ab)" is a problem of involution. Since a power is a special kind of product, involution may be effected by repeated multiplication.

216. Law of Signs. According to § 50.

+a+a+a+a3.

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1. All powers of a positive quantity are positive.
2. All even powers of a negative quantity are positive.
3. All odd powers of a negative quantity are negative.
(-a) is positive, (— ab2)9 is negative.

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4. (-3a2b3) 4 = (-3a2b3) · (-3a2b3)·(-3a2b3)·(-3a2b3)

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To find the exponent of the power of a power, multiply the given exponents.

To raise a product to a given power, raise each of its factors to the required power.

To raise a fraction to a power, raise its terms to the required power.

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INVOLUTION OF BINOMIALS

218. The square of a binomial was discussed in § 65.

(a+b)2= a2+2ab+b2.

219. The cube of a binomial we obtain by multiplying (a+b)3 (a+b)3 = a3 +3 a2b+3 ab2 + 63,

by a+b.

and

(a - b)3 a3-3 a b+3ab2-63.

=

Ex. 1. Find the cube of 2x+3y.

(2x+3y)3 = (2x)3 + 3(2 x)2 (3 y) + 3(2 x) (3 y)2 +(3 y)8
= 8 x3 +36 x2y + 54 xy2 + 27 y3.

Ex. 2. Find the cube of 3 x2 — y".

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220. The higher powers of binomials, frequently called expan

sions, are obtained by multiplication, as follows:

(a+b)2= a2+2ab+b2.

(a + b)3 = a3 + 3 a2b + 3 ab2 + b3.

(a+b)*= a* +4 a3b + 6 a2b2+4 ab3 + b2.

(a+b)5 = a + 5 a1b+10 a3b2 + 10 a2b3 +5 ab1 + b3, etc.

An examination of these results shows that:

1. The number of terms is one greater than the exponent of the binomial.

2. The exponent of a in the first term is the same as the exponent of the binomial, and decreases in each succeeding term by one. 3. The exponent of b is 1 in the second term of the result, and increases by 1 in each succeeding term.

4. The coefficient of the first term is 1.

5. The coefficient of the second term equals the exponent of the binomial.

6. The coefficient of any term of the power multiplied by the exponent of a, and the result divided by 1 plus the exponent of b, is the coefficient of the next term.

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(x − y)5 = x5 + 5 x1 ( − y) + 10 x3 (− y)2 + 10 x2( − y)3 + 5 x( − y)4

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221. The signs of the last answer are alternately plus and minus, since the even powers of y are positive, and the odd -y powers negative.

Ex. 3. Expand (2x-3y3).

(2 x2 — 3 y3)1 = (2 x2)1 — 4 (2 x2)3 (3 y3) +6 (2 x2)2 (3 y3)2

-4(2x2) (3 y3) + (3 y3)*

= 16x8-96xy3+216 x1y-216 x2y+81 y12.

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222. The square of polynomials was discussed in § 67.

223. The higher powers of polynomials are found either by multiplication, or by transforming the polynomials into binomials.

Ex. 1. Expand (a+b-c)3.

(a+b−c)3=[(a+b)−c]3

=(a+b)3-3(a+b)2c+3(a+b)c2-c8

=a3+3 a2b+3 ab2+b3−3 c(a2+2 ab+b2)+3 ac2+3 bc2 ....-c3

=a3+3 a2b+3 ab2+b3 −3 a2c-6 abc-3 b2c+3 ac2+3 bc2-c3.

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