4. 5abc+14 m2 - (10 a b c -3 m2-3 cd). . 5.-7 ax2 y2+3x2y — (15x2y-10 x2 y2-5abc). 6. a2+6ax +10 a x2-(a2-6 ax-10 ax2). 7. m1-(a2-2 m3). 8. 7xz-3xz2+3x3-(3 x 2-7 x z + 10 z3). 9. 4x2 y2-4 x y3-1-(2 x2 y2—4 x y3 +5). 10 30 a b 15 ac2 + 15 m2 x (15 a c2+30 a b — 15 m2 x). ART. 40. It is often useful to reverse the process in the last article, and place part of a polynomial within a parenthesis, preceded by the sign —. This may be done without altering the value of the polynomial, provided the signs of all the terms placed within the parenthesis are changed. Thus, a-m+n may be written a- (m—n); for if the subtraction indicated in the latter expression be performed, we obtain a―m+n. Let the student throw all, except the first term, of each of the following polynomials into a parenthesis, preceded by the sign 1. m2-ca. 2. 4a2+b-3 c2. 3. 150-2x+4y-7ab. 4. 6x2 y2+2ax+3by-10 z2. 5. a2 m2-10+6xy. 6. 3abcmx-3y+10. 7. 4x2 y3-6x y3 + a b c — m2 x. 8. 20 p2q-56+x2. 9. a2 b2c2+10-a3bc2abc. SECTION XI. 4 MULTIPLICATION OF POLYNOMIALS. ART. 41. Let it be required to multiply 7+3 by 5. Since 73 is 10, the product must be 5. 10, or 50. Now, to multiply without reducing the multiplicand, it is evident that we must multiply both 7 and 3 by 5, and add the products. In like manner, to multiply a+b by c, we must multiply both a and b by c, and add the products. But if any term of the multiplicand has the sign, the monomial multiplier being affected with the sign +, the corresponding product must have the sign — As an example in numbers, let us multiply 9-5 by 3. Since 9-5 is 4, the product must be 3.4, or 12. But if we first multiply 9 by 3, the product 27 is too great by 3 times 5, or 15; we must, therefore, multiply 5 by 3, and subtract the product 15 from 27, which gives 2715, or 12. Operation. 3 Product 27 — 15 = 12. b by c, if we. first In a similar manner, to multiply a multiply a by c, the product a c is too great by c times b; we must, therefore, multiply b by c, and subtract the product b c from a c, which gives a c-bc for the corre result. Hence we see that −b, multiplied by +c, gives — b c If both of the quantities whose product is sought are polynomials, the whole of the multiplicand must be multiplied by each term of the multiplier. Let it be required to multiply 7+2 by 4+3. Since 72 is 9, and 4+3 is 7, the product must be 7.9, or 63. To produce this, 7+2 must first be multiplied by 4, which gives 28+8; then 7-2 must be multiplied by 3, which gives 21+6; the sum of these products is 28 +8+21+6, or 63. Operation. 7+2 Product 28+8+21+6=63. In a similar manner, to multiply a+b by c+d, we first multiply a+b by c, which gives ac+bc; then multiply a+b by d, which gives ad+bd; now, adding these products, we have a c+be+ad+bd for the entire product. Operation. Now, let it be required to multiply 6-2 by 5—3. Since 6 2 is 4, and 5-3 is 2, the product must be 4.2, or 8. If we first multiply 6-2 by 5, we have 30 10, or 20; this is too great by 3 times 6 18-6, or 12; we must, therefore, subtract 18-6 from 30-10, which gives 30-10-18+6, or 8. In like manner, to multiply a b by c-d, we first multiply ab by c, and the product, as has already been seen, is a c—bc; this is too great by d times a—b, which is a d―bd; this last product must therefore be subtracted from a c-bc, which gives a c -bc-ad+ bd, for the true product of a-b by c -d Operation. с d Product ac· ·bc-ad+bd. On examining the preceding product, we see that ad was produced by multiplying + a by-d; hence, if a term having the sign + be multiplied by a term having the sign, the corresponding product must have the sign. Also, bd was produced by multiplying — b byd; hence, if a term having the sign be multiplied by another term having the sign, the product must have the sign; in other words, if two negative terms are multiplied together, the product must be positive. ART. 42. From the preceding analysis, we derive the following RULE FOR THE MULTIPLICATION OF POLYNOMIALS. 1. Multiply all the terms of the multiplicand by each term of the multiplier separately, according to the rule for the multiplication of monomials. 2. With regard to the signs, observe, that if the two terms to be multiplied together have the same sign, either both, or both, the corresponding product must have the sign +; but if one term has the sign +, and the other the sign, the corresponding product must have the sign 3. Add together the several partial products, reducing terms which are similar. 1 Multiply 4 a x + 7 c 2 y by 2 ax+5c2 y 8 a2x2+14 ac2 x y Partial +20 ac2 x y +35 c4 y2 products. 8 a2x2+34 a c2 xy +35 c4 y2; the entire product reduced. To facilitate reduction, it is advisable to place the similar terms of the partial products under each other. -5a2 b2c-15 62 c2+5bc3 products. b2 2 a4 b2-2 a2 b2 c-a2 b c2-15 62 c2+5bc3. Result. 3. Multiply x2+x3 y + x2 y2+x y3 + y2, by X- -y x5+x1 y + x3 y2 + x2 y3 + x y1 Partial -x1y — x3 y2 — x2 y3 — x y1 —-y5 products. x5-y5. Result. |