Product x2+4x3y—60x2y2+11x2-2xy+28. 2 9. Multiply 3x2y+xy2+y3 by x-y. Ans. x. 10. Multiply x2+xy+y2 by x2-xy+y3. Ans. x2+x2y2+ya. 11. Multiply 3x2-2xy+5 by x2+2xy—3. Ans. 3x4x34x2-4x2y2+16xy-15 12. Multiply 2a-3ax+4x2 by 5a2-6x-2x2. Ans. 10a427a3x+34a2x2-18ax3-8x*. DIVISION. DIVISION in Algebra, as well as in Arithmetic, is the converse of multiplication, and is performed by beginning at the left hand, and dividing all the parts of the dividend by the divisor, when it can be done; or by setting them down like a vulgar fraction, the dividend over the divisor, and then reducing the fraction to its lowest terms. In division the rule for the signs is the same as in multiplication, viz. if the signs of the divisor and dividend be alike, alike, that is, both or both, then the sign of the ; but if they be unlike, the sign of CASE I. When the divisor and dividend are both simple quantities. RULE. 1. Place the dividend above a line, and the divisor under it, in the form of a vulgar fraction. 2. Expunge those letters, that are common to the dividend and divisor, and divide the coefficients of all the terms by any number, that will divide them without a remainder, and the result will be the quotient required. *Because the divisor, multiplied by the quotient, must produce the dividend. Therefore, 1. When both the terms are +; the quotient must be +, because in the divisor X + in the quotient produces + in the dividend. ; the quotient is also +, be 2. When the terms are both cause- in the divisor X + in the quotient produces dividend. 3. When It may not be amiss to observe, that when any quantity is divided by itself, the quotient will be unity, or 1; because any thing contains itself once: thus xx gives 1, and ✔✅2ab divided by ✔✅2ab¦ gives 1. NOTE 1. To divide any power by another of the same root; subtract the exponent of the divisor from that of the dividend, and the remainder will be the exponent of the quotient. Thus, 3. When one term is + and the other ; the quotient must be -, because in the divisor X in the quotient produces in the dividend; or in the divisor X + in the quotient gives So that the rule is general; like signs give +, and unlike signs give, in the quotient. Thus, the quotient of a divided by a3 is a-3, or as. That of " by x" is x". And that of x" by x" is x”-”. But it is to be observed, that when the exponent of the divisor is greater than that of the dividend, the quotient will have a negative exponent. Thus, the quotient of x3 divided by x7 is 5-7, or x-2. And that of ax' by x5 is ax-3. -3 And these quotients, viz. x and ax-3, are respective In like manner, ax" divided by cx1" gives m n And the quotient of aa+x2 1′′ divided by a2+x1Ï”. is、 And ab-+**1" divided by ab+x1" gives ab+x2]" SCHOLIUM. SCHOLIUM. When fractional exponents of the powers of the same root have not the same denominator, they may be brought to a common denominator, like vulgar fractions, and then their numerators may be added, or subtracted, as before. ac+x] Thus, the quotient of ac+x divided by ac+x ac+x | ÷ = ac+x} & = ac+xl. is NOTE 2. Surd quantities under the same radical sign are divided, one by the other, like rational quantities, only the quotient, if it do not become rational, must stand under the same radical sign. Thus, the quotient of ✔✅/21 divided by √3 is 7. When the divisor is a simple quantity and the dividend a com pound quantity. RULE. Divide every term of the dividend by the divisor, as in the first case. N N EXAMPLES. |