+

-4, x= −3, x=−2, x= −1, x=0, x=1, x=2, x=3, z=4

Hence, we conclude that two of the roots lie between 0 and 1; one between 3 and 4; and one between 3 and 4. Here, we have found the values of the roots, to within less than 1. The method of completing the approximation, will be explained hereafter.

In the preceding case, we have seen that the same number of permanences, as

;

4 gives hence, no

real roots lie between + 4 and +∞. We have also seen, that 4 and

∞, give the same

number of variations; hence, no real root lies between them. The values, 4 and 4, are called the LIMITS of the roots of the given equation; the former being the inferior, and the latter, the superior limit.

If we consider the positive roots alone, 0 and 4 are the limits. If we consider the negative roots alone, - 4 and 3 are the limits. In the same manner, the limits of the positive and negative roots of any equa tion may be found. It is often useful to determine these limits, especially when seeking the entire roots of an equation, by the process of Article 204.

EXAMPLES.

1. Find the number, the places, and the limits of the real roots of the equation, x3 + x2 + x

100 = 0.

X3

=

·∞,x=+∞, x=0, x=1, x=2, x=3, x=4, x=5,

Hence, there is one real root lying between 4 and 5, which are the limits of the root.

2. Find the number, places, and limits of the real roots of the equation,

x48x3 + 14x2 + 4x 8 0.

Ans. There are 4 real roots; one between 0 and 1, one between 2 and 3, one between 5 and 6, and one between 1 and 0. The limits are 1 and 6.

3. Find the number, places, and limits of the real roots of the equation, x3 = 0.

23x 24

Ans. 3 real roots, .one between 5 and 6, one between

1 and 2, and one between

4 and

5. The

4. Find the number, places, and limits of the real

Ans. There is 1 real root, and it lies between the

limits 1 and 2.

APPENDIX.

1. Ir is proposed to demonstrate, in this Appendix, several useful principles; such as those employed in fac toring, the Binomial Theorem for any exponent, the method of summing Series, and the like. These principles are all of high importance in the more advanced portions of the mathematical course; but, on account of their difficulty, it has not been thought best to introduce their demonstration into the body of the work.

Principles employed in Factoring.

I. The difference of the like powers of any two quantities, is always divisible by the difference of the quantities.

2. To demonstrate this principle, let a and b denote any two quantities whatever, and m any positive whole number whatever; then will am bm denote the differ. ence between the like powers of any two quantities, and α b the difference between the quantities: If we commence the division by the rule, we shall have the following operation :

The remainder may be factored, and, placed under the

The second member of Equation (1) will be entire, and consequently, the first member will be entire, when

απι

-1

bm-1

is entire that is, if the difference of the a b (m1)th powers of two quantitise is divisible by the difference of the quantities, then will the difference of the mth powers of the two quantities also be divisible by the difference of the quantities.

But, we know that the difference of the second powers is divisible by the difference of the quantities, giving for a quotient the sum of the quantities; hence, from the principle above demonstrated, the difference of the cubes is also divisible by the difference of the quantities. It having been proved that the difference of the cubes is divisible, it follows, from the principle demonstrated, that the difference of the fourth powers is also divisible by the difference of the quantities. The difference of the fourth powers being divisible, it follows, as before, that the dif ference of the fifth powers is divisible; and so on, by successive deduction, it may be shown that the division. is possible when m is any positive number whatever. Hence, the principle is proved.

We found the first term of the quotient to be, am-1; and if we perform a second partial division, we shall get, for the second term of the quotient, am-2b, with a sec ond remainder, b2(am--2 — ¿m~~2); dividing again, we shali

gct, for a third term of the quotient, am-362; and so on. We see that the exponent of a goes on diminishing by 1, in each term of the quotient, to the right; whilst the exponent of b, being 0, in the first term, goes on increasing by 1, in each term, to the right: the last tcrm of the quotient contains a to the 0 power, and b to the Hence, we may write the quotient as

m

1 power.

1 = am―1+am−2b+am−3b2+ ... + a2bm-3+abm-2

+bm-1

(2.)

If, in Equation (2), we replace a by c2, and b bv d2, we shall have,

Whatever may be the value of m, 2m will be an even number, and the second member of (3) will be entire; hence, we conclude that,

II. The difference of like even powers of any two quantities is always divisible by the difference of the squares of the quantities.

3. If we multiply both members of (3), by (c – d), we shall have

(c−d) (c2m−2+ c2m−4 d2 + +c2d2m−4+d2m−2)

(4.)

The second member of (4) is entire, consequently the first member is also entire. Hence,

III. The difference of like even powers of any two quantities, is always divisible by the sum of the quantities.