SECTION III.

INVOLUTION.

CHAPTER I.

(Art. 64.) Equations, and the resolution of problems producing equations, do not always result in the first powers of the unknown terms, but different powers are frequently involved, and therefore it is necessary to investigate methods of resolving equations containing higher powers than the first; and preparatory to this we must learn involution and evolution of algebraic quantities.

(Art. 65.) Involution is the method of raising any quantity to a given power. Evolution is the reverse of involution, and is the method of determining what quantity raised to a proposed power will produce a given quantity.

As in arithmetic, involution is performed by multiplication, and evolution by the extraction of roots.

The first power is the root or quantity itself;

The second power, commonly called the square, is the quantity multiplied by itself;

The third power is the product of the second power by the quantity;

The fourth power is the third power multiplied into the quantity, &c.

The nth power of a1 has the exponent 4 repeated n times, or a. Therefore, to raise a simple literal quantity to any power, multiply its exponent by the index of the required power.

Raise to the 5th power. The exponent is 1 understood, and this 1 multiplied by 5 gives x5 for the 5th power.

(Art. 66.) By the definition of powers the second power is any quantity multiplied by itself; hence the second power of ax is a2x2, the second power of the coefficient a, as well as the other quantity x; but a may be a numeral, as 6x, and its second power is 36x2. Hence, to raise any simple quantity to any power, raise the numeral coefficient, as in arithmetic, to the required power, and annex the powers of the given literal quantities.

EXAMPLES.

1. Required the 3d power of 3ax2. 2. Required the 4th power of y2. 3. Required the 3d power of -2x. 4. Required the 4th power of -3x.

Ans. 27a3x6.

Ans. y. Ans. -8x3.

Ans. 81x4.

Observe, that by the rules laid down for multiplication, the even powers of minus quantities must be plus, and the odd powers minus.

(Art. 67.) The powers of compound quantities are raised by

the mere application of the rule for compound multiplication. (Art. 12.)

Let a+b be raised to the 2d, 3d, 4th, &c. powers

By inspecting the result of each product, we may arrive at general principles, according to which any power of a binomial may be expressed, without the labor of actual multiplication. This theorem for abbreviating powers, and its general application to both powers and roots, first shown by Sir Isaac Newton, has given it the name of Newton's binomial, or the binomial theorem.

OBSERVATIONS.

Observe the 5th power: a, being the first, is called the leading term; and b, the second, is called the following term. The sum

of the exponents of the two letters in each and all of the terms amount to the index of the power. In the 5th power, the sum

of the exponents of a and b is 5; in the 4th power it is 4; in the 10th power it would be 10, &c. In the 2d power there are three terms; in the 3d power there are 4 terms; in the 4th power there are 5 terms; always one more term than the index of the power denotes.

The 2d letter does not appear in the first term; the 1st letter does not appear in the last term.

The highest power of the leading term is the index of the given power, and the powers of that letter decrease by one from term to term. The second letter appears in the 2d term, and its exponent increases by one from term to term as the exponent of the other letter decreases.

The 8th power of (a+b) is indicated thus: (a+b). When expanded, its literal part, (according to the preceding observations) must commence with a3, and the sum of the exponents of every term amount to 8, and they will stand thus: a3, a1b, aob2, ab3, a1b1, a3b3, a2bo, ab1, bo.

The coefficients are not so obvious. However, we observe that the coefficients of the first and last terms must be unity. The coefficients of the terms next to the first and last are equal, and the same as the index of the power. The coefficients increase to the middle of the series, and then decrease in the same manner, and it is manifest that there must be some law of connection between the exponents and the coefficients.

By inspecting the 5th power of a+b, we find that the 2d coefficient is 5, and the 3d is 10.

The third coefficient is the 2d, multiplied by the exponent of the leading letter, and divided by the exponent of the second letter increased by unity.

In the same manner, the fourth coefficient is the third multiplied by the exponent of the leading letter, and divided by the exponent of the second letter increased by unity, and so on from coefficient to coefficient.

K

Now as the exponents of a and b are equal, we have arrived at the middle of the power, and of course to the highest coefficient. The coefficients now decrease in the reverse order which they increased.

Hence the expanded power is

a3+8ab+28aob2+56a3b3+70aaba+56a3b3+28a2b®+8ab2+68.

Let the reader observe, that the exponent of b, increased by unity is always equal to the number of terms from the beginning, or from the left of the power. Thus, b2 is in the 3d term, &c. Therefore in finding the coefficients we may divide by the number of terms already written, in place of the exponents of the second term increased by unity.

If the binomial (a+b) becomes (a+1,) that is, when b becomes unity, the 8th power becomes,

a3+8a2+28a®+56a3+70a1+56a3+28a2+8a+1.

Any power of 1 is 1, and 1 as a factor never appears.
If a becomes 1, then the expanded power becomes,

1+86 +2862+56b3+70b1+56b5+2866+8b7+b9