From the preceding rules are deduced the following useful theorems, viz: 1. By the rule for addition, if the sum of any two quantities, a and b, be added to their difference, the sum will be twice the greater.* 2. By the rule for subtraction, if the difference of any two quantities be taken from their sum, the remainder will be twice the less.† 3. By multiplication, example 2, article 21, if the sum of any two quantities be multiplied by their difference, the product will be the difference of their squares. Now, it is easy to perceive that the next, or sixth term of the quotient will be 25 and the seventh term and so on, alternately plus and minus; this is called the law of continuation of the series. And the sum of all the terms, when infinitely continued, is said to be equal to the fraction Thus we say the 2 a a+x vulgar fraction when reduced to a decimal, is .66666, &c. 3' infinitely continued. The terms in the quotient are found by dividing the remainders by a, the first term of the divisor; thus, the first remainder, -x, divided by a, gives -- a the quotient; and the second remainder,+ the second term in EXAMPLES FOR PRACTICE. Divide x 48x+200 by x+2. Ans. x-50x+100. 2. Divide 3-22x-24 by x+4. Ans. x2-4x-6. 3. Divide +9x2+4x-80 by x+-5. Ans. x2-4x—16. 4. Divide +39 +249x+289 by x+1. Ans. x2-40x+289. 38x+2102538x+289 by x+1. 5. Divide x b b2 b3 b4 Answer 23-39x2+249x+289. a±l) a (1±±, &c. answer. a3 263 -2ab-4b2 6. Divide x+6x1—10x3-112x2--207x-110 by x+5. 7. Divide Answer +-15x2-37x-22. +3-15x2-37x-22 by x+2. Answer x--x2-13x-11, and x2-2x-11. 8. Divide x3-x2-13x-11 by x+1. Ans. 2—2x—11. 9. Divide 2+6x-10x3-112x2-207x-110 by x+2. 10. Divide 11. Divide, Ans. x+4x3-18x2—76x—55. 4x3-18x2--76x-55 by x+5. Ans. x3-x-13x-11. x+6x-10x3-112x2-207x--110 by x+1. 12. Divide 2+5x3-15x2-97x--110 by x+2 and by x+-5. 13. Divide Ans. x3-3x2-21x-55. 219x+123x-302x-200 by 2-4. Ans. -15x+63x+50. 14. Divide x1-27x3+262x2+356x-1200 by x+3. Answer 2-30x2+252x-400. ON THE REDUCTION OF ALGEBRAIC FRACTIONS. (42.) The rules managing algebraic fractions being of the same nature as vulgar fractions in common arithmetic, the operations are performed exactly in the same manner. CASE I. To reduce a mixed quantity to an improper fraction. RULE. Multiply the integer, or whole part, by the denominator of the fraction, and to the product add the numerator; then under their sum place the original denominator. CASE II. To reduce an improper fraction to a whole, or mixed quantity. This is the same as Case II. in division. RULE. Divide the numerator by the nominator for the integral part; and if there be a remainder, place it over the denominator for the fractional part, with the proper sign prefixed. 10 Ans. 2x-1+ Ans. 5x 5x |