Product 4x+20x1y+25y* (60.) The examples hitherto given in multiplication have been confined to positive quantities, and the products have all been positive. We must now establish a general rule for the signs of the product. First. If +a is to be multiplied by +b, this signifies that +a is to be repeated as many times as there are units in b, and the result is +ab. That is, a plus quantity multiplied by a plus quantity gives a plus result. Secondly. If a is to be multiplied by +b, this signifies that a is to be repeated as many times as there are units in b. Now -a, taken twice, is obviously -2a, taken three times is -3a, etc.; hence, if -a is repeated b times, it will make -ba, or —ab. That is, a minus quantity multiplied by a plus quan. tity gives minus. Thirdly. To determine the sign of the product when the multiplier is a minus quantity, let it be proposed to multiply 8-5 by 6-2. By this we understand that the quantity 8-5 is to be repeated as many times as there are units in 6-2. If we multiply 8-5 by 6, What is the product QUEST.-What is the product of +a by +b? ofa by +b? When the multiplier is a minus quantity? we obtain 48-30; that is, we have repeated 8-5 six times. But it was only required to repeat the multiplicand four times, or (6-2). We must therefore diminish this product by twice (8—5), which is 16–10; and this subtraction is performed by changing the signs of the subtrahend; hence we have 48-30-16+10, which is equal to 12. This result is obviously correct, for 8-5 is equal to 3, and 6-2 is equal to 4; that is, it was required to multiply 3 by 4, the result of which is 12, as found above. (61.) In order to generalize this reasoning, let it be proposed to multiply a-b by c-d. If we multiply a-b by c, we obtain ac-bc. But it was proposed to take a-b only as many times as there are units in the difference between c and d; therefore the product ac-bc is too large by a-b taken d times; that is, to have the true product, we must subtract d times a-b from ac-bc. But d times a-b is equal to ad-bd, which subtracted from ac-bc, gives ac-bc-ad+bd. Thus we see that a multiplied by -d gives ―ad, and -b multiplied by -d gives +bd. Hence a plus quantity multiplied by a minus quantity gives minus ; and a minus quantity multiplied by a minus quantity gives plus. (62.) The preceding results may be briefly express ed as follows: + multiplied by +, and — multiplied by —, give +. + multiplied by —, and — multiplied by +, give QUEST.-What is the product of a-b by c-d? How results be expressed! may these Or, the product of two quantities having the same sign, has the sign plus; the product of two quantities having different signs, has the sign minus. (63.) Hence all the cases of multiplication are comprehended in the following RULE. Multiply each term of the multiplicand by each term of the multiplier, and add together all the partial products, observing that like signs require + in the product, and unlike signs Ex. 7. Multiply 6a+12ax-6x' by 2a-4ax-2x2. Ans. 12a-72a2x2+12x1. Ex. 8. Multiply x2-2xy-3 by 5x2+10xy+15. Ans. 5x-20x3y'—60xy-45. Ex. 9. Multiply 2x2+2x3y2+2y3 by 3x3-3x2y2—3y3. Ans. 6x-6x'y'—12x'y'—6y'. Ex. 10. Multiply a'-2b'+c' by a'-b'. Ans. Ex. 11. Multiply 5a-2ab+4a2b1 by a3-4a3b+2b3. QUEST.-Give the general rule for multiplication. Ex. 12. Multiply 4a3-5a3b-8ab2+2b3 by 2a2-3ab -46'. Ans. 8a-22a'b-17a b'+48a2b3+26ab*— 8b3. Ex. 13. Multiply 3a2-5bd+ef by −5a2+4bd-8ef. Ans. -15a+37abd-29aef-20b'd'+44bdef-8e1ƒ3. Ex. 14. Multiply x*+2x2+3x2+2x+1 by x2-2x+1. Ans. x-2x2+1. Ex. 15. Multiply 14a3c-6a2bc+c2 by 14a3c+6a2bc -c2. Ans. Ex. 16. Multiply 3a3+35a3b-17ab2-13b3 by 3a1 +26ab-57b2. Ans. (64.) For many purposes it is sufficient merely to indicate the multiplication of two polynomials, without actually performing the operation. This is effected by inclosing the quantities in parentheses, and writing them in succession, with or without the interposition of any sign. Thus (a+b+c) (d+e+ƒ) signifies that the sum of a, b, and c is to be multiplied by the sum of d, e, and ƒ. When the multiplication is actually performed, the expression is said to be expanded. (65.) The following theorems are of such extensive application that they should be carefully committed to memory. THEOREM I. The square of the sum of two quantities is equal to the square of the first, plus twice the product of the first by the second, plus the square of the second. QUEST.-HOW may we indicate the multiplication of polynomials? What is the square of the sum of two quantities? we obtain the product a2+2ab+b2. Hence, if we wish to obtain the square of a binomial, we can write out the terms of the result at once, according to this theorem, without the necessity of performing an actual multiplication. Examples. Ex. 1. (2a+b)2=4a2+4ab+b2. This theorem deserves particular attention, for one of the most common mistakes of beginners is to call the square of a+b equal to a2+b2. THEOREM II. (66.) The square of the difference of two quantities is equal to the square of the first, minus twice the product of the first and second, plus the square of the second. Thus, if we multiply by a ·b a b we obtain the product a2-2ab+b3. QUEST.-Illustrate by an example. What mistake do beginners fre quently commit? What is the square of the difference of two quanti ties? |