4, 5 and 6

6, 3 and 9

8, 12 and 10

20, 30 and 40

4. Given, (x-a) (x —b) (x + c) = 0.

Ans. xa, x = = b, x= - C. Multiplying the factors as indicated, we have

x3 — (a+b—c) x2+(ab—a c—bc)x+abc=0 In this example, one of the roots is minus or negative, and it will be seen by inspecting the equation, that the remarks made on the formation of the equation from the roots, in problem 1, are applicable here; provided, we understand by the sum of the roots, sum of the products, &c., the result of uniting them according to their signs. The sum of a, b and -C is the result of adding a and b, and subtracting c from their sum; the sum of ab- — a c and — bc is the result of subtracting ac and b c from a b, &c.; and it will be seen by inspecting the following equations, that the same remarks are applicable when two, or all of the roots are negative.

(x-a)(x+b) (x + c) = 0 Orx3-(a—b—c) x2+(—ab—a c + b c) x — a b c = 0 What equations will have the following roots?

Discussion of Equations of the Fourth and Fifth Degrees. If (1) (x-α) (x — b) (x — c) (x —d)=0; what are the values of x?

=+d.

Ans. x=+ α, x = +b, x=+c, x = Multiplying the factors in the first member of the preceding equation, we have

(2) x1— (a+b+c+d) x2 + (a b+ac+ad+bc+b d+ cd) x2-(abc+abd+acd+bed)x+abed=0

This equation is the same as equation (1,) and would give the same values of x. It is called a biquadratic equation, or an equation of the fourth degree.

K

1st. We see that an equation of the fourth degree gives four values for x.

2nd. That if the first term is once the fourth power of the unknown quantity, that the coefficient of the second term is the sum of all the roots, or values of x, taken with the contrary sign.

3rd. That the coefficient of the third term, is the sum of all the products which can be made, by taking the roots, two by

two.

4th. That the coefficient of the fourth term is the sum of all the products that can be made, by taking the roots, three by three, with the sign changed.

5th. That the fifth term is the continued product of all the

roots.

What are the values of x in the following equation?

(x+a) (x + b)(x — c) (x — d) = 0

Ans. x=- а, x= -b, x =

+c, x = +d. Multiplying the factors as in the preceding, we have x2-(—a—b + c + d) x2 +(ab-ac-ad-bc-bd+ cd) x2-(abc+abd-acd-bcd)x+abcd=0

It will be seen by inspecting the preceding equation, that the same remarks, as in the preceding case, are applicable. If we understand by the sum of the roots, the sum of the products, &c., the roots and products united according to their signs.

Form the equations which shall have the following roots: +4, +5, + 6 and + 12

(1)

Find the values of x in the following equation:

(x — a) (x —— b) (x — c) (x — d ) (x — e) (x — ƒ) = 0 What are the values of x?

If the factors in the preceding equation be multiplied together, what will be the form of the equation? Of what degree is it? How many roots has it? What is the first term of the equation? What is the second, third, &c.?

If x33x

-360, or x3 +0 x2 + 3 x −36=0; what is the sum of the roots? What is their continued product? When the second term of an equation, or the term contain

ing the power of x next the highest is wanting; what is the sum of the roots?

When the last term of a cubic equation is minus; how many of the roots are positive?

When all the roots of an equation are positive; what are the signs of the terms?

What equations have the following roots?

(1)

=

SECTION LIII.

General Discussion of Equations.

The equation (-a) (x—b) (x —c) (x-d) = 0, may be divided by x- -a, and we shall have (x—b) (x — c) (x — d) 0, an equation of the third degree. Divide this last equation by x-b, and we have (x-c) (x— d) = 0, an equation of the second degree. Divide this equation by x c, and we

have

d=0.

In this manner, any equation may be reduced one degree lower, if one value of x is known, by bringing all the terms into the first number, and dividing by the x united to its known value, with the sign changed.

Given, (1) x -10 x3 +35 x2

50x24 0; and one of the values of x = 1. It is required to reduce the equation one degree lower.

x-1)x4103 +35 x2-50 x + 24 (x2-9 x2+26x-24 X4 ემ

Given +2, one root of this equation, to reduce the equation one degree.

x—2) x3-9 x2+26x-24 (x2-7x+12

2x2

Given, one root of this equation 3, to reduce it one degree. x-3) x2-7x+12 (x-4

To find all the values of x, then, in an equation of any degree: Find first, one value by trial, or by one of the preceding rules; then reduce the equation one degree, by division; and find a value of x in the equation reduced; then the equation may be reduced again, one degree, by division, and another value of x found, &c., until all the values of ≈ are obtained.

When the equation is reduced to the second degree, the two remaining values of x may be found by the common rule for affected equations.

When any equation is reduced to one of the second degree, both the values remaining may be imaginary; and we have seen, that if one is imaginary, the other must be.

Can any equation have one imaginary root, and only one? Can any equation of an odd degree have all its roots imaginary?

Find all the values of x in the following equations:

SECTION LIV..

Arithmetical Progression.

A series of terms increasing or decreasing by a common difference, is called an arithmetical progression. 1, 2, 3, 4, 5, 6. 1, 3, 5, 7, 9. 18, 15, 12, 9, 6, &c. are three series in arithmetical progression. The common difference in the first is 1, in the second 2, and in the third 3.. The first two are called increasing series, the last, a decreasing series.

1. If a represents the first term, and r the common difference; what will represent the second term of an increasing series? Ans. ar. What the third? What the fourth? &c.

2. Any term of such a series is always equal to the first term, added to how many times the common difference?

3. What will represent the xth term of the series 3, 5, 7, 9, 11, &c. Ans. 3+2(x-1,) or 3 + 2 x — 2.

4. What will represent the xth terms of the following series: 4, 6, 8, 10, &c.? 10, 13, 16, 19, &c.? 14, 18, 22, &c.? 1 3 5 7. &c.?

4 4 4 49

5. What will represent the xth term of the series a, a +r, a + 2 r, a + 3 r, &c.? Ans. a+ r (x— 1,) or a + xr — r. 6. What will represent the xth term of the series 1, 2, 3, 4, 5, &c.?

7. A man set out on a journey, traveling 6 the first hour, 6 the second, 63 the third, &c. At length he found that he had traveled as many miles the last hour, as he had traveled hours. How many hours had he traveled?

8. Two men set out from two towns with a design to meet. One traveled 1 mile the first hour, 3 the second, 5 the third, &c. The other traveled 3 miles the first hour, 4 the second, 5 the third, &c. When they met, they found that the number of miles they had traveled the last hour, was 10 less than 4 times the number of hours. How many hours had they traveled?

9. Two men, A and B,,owed some money.. A paid every day 10 dollars. B paid the first day 2, the second 21, the third 3, &c. On a certain day it was found that the number of dollars B paid, multiplied by the number of payments he