When a parenthesis with the sign minus before it is removed, the sign of each term within the parenthesis must be changed according to the Rule for subtraction. Thus, 3 3 (x‹3 + y3 — 23) = x3 — z3 — x3 — y3 + And conversely, A polynomial, or any number of the terms of a polynomial, can be enclosed in a parenthesis and the minus sign placed before the parenthesis without changing the value of the expression, providing the signs of all the terms are changed from plus to minus or from minus to plus. Thus, a2 — b2 + c2 + d − x = a2 — (b2 — c2 — d + x). NOTE. When the sign of the first term in the parenthesis is plus, the sign need not be written. (Art. 18.) According to this principle a polynomial can be written in a variety of ways. Thus, x3-3x2y+3 x y2—y3—x3 — (3 x2 y — 3 x y2+y3) =x3-3x2y-(-3xy2+y3) = x2+3 x y2 (3x2y + y3) =x3 — y3 — (3x2y — 3xy2) &c. - Remove the parenthesis, and reduce each of the following examples to its simplest form. 2. x2 — 6 a x + x3 — 6 x2 y — (x2 + 6 a x + x2-6x2 y). x2 Ans. 12 ax. 3. m2 — n2 + 2 x (4 m2 + 3 n2 - 4 c). 4. 16xy+14c18y-(-14c+27 y 16 x y). 1. The last three terms of 7 x2-14 xy-3x+4y. Ans. 72(14 xy+3z-4 y). 2. The last three terms of x2 + y2 — 3 x y + 4 c. 3. The last four terms of 4 a 7b 6 c 4. The last four terms of a2+ b2 + c2 — d2 + a3. 5. Write in as many forms as possible by enclosing two or more of the terms in parenthesis, a3 — b3 + c3 45. In subtraction, when two quantities have a common factor their difference is the difference of the coefficients of the common factor multiplied by this factor. 3. From a x3 take bx3 b x2. Ans. (a - b) x3 + b x2. 4. From 4 x2 6x take a x2 + bx. Ans. (4a) x2 - (6 + b) x. 5. From 6a+ 4 a2 — a take a3 x — a2 y + a z. Ans. (6x) a3+ (4 + y) a2 − (1 + z) a. 6. From ab b c take 3b+cx. -- 7. From a2- bx + c√x take b x2 + c x − d√√x. 8. From xy + x2- x2 y2 take y2+ x2y — x2 y2. SECTION VI. MULTIPLICATION. 46. MULTIPLICATION is a short method of finding the sum of the repetitions of a quantity. 47. The multiplier must always be an abstract number, and the product is always of the same nature as the multiplicand. The cost of 4 pounds of sugar at 17 cents a pound is 17 cents taken, not 4 pounds times, but 4 times; and the product is of the same denomination as the multiplicand 17, viz. cents. In Algebra the sign of the multiplier shows whether the repetitions are to be added or subtracted. 1. (a) x (+4)=+4a; i. e. a added 4 times is +a+a+a+a+4 a. i. e. a subtracted 4 times is +a+a+a+a+4 a. In the first and second examples the nature of the product is; in the first, the the product is to be added, and sign of 4 sign of 4 shows that 4a added is + 4 a; shows that the product in the second, the 4 a. In the third and fourth examples the nature of the product is ; in the third, the sign of 4 shows that the product is to be added, and 4 a added is 4a; in the fourth, the sign of 4 shows that the product is to be subtracted, and 4 a subtracted is + 4 a. 48. Hence in multiplication we have for the sign of the product the following RULE. Like signs give +; unlike, -. Hence the products of an even number of negative factors is positive, of an odd number, negative. 49. Multiplication in Algebra can be presented best under three cases. CASE I. 50. When both factors are monomials. 1. Multiply 3 a by 2b. OPERATION. 3 a X 2b= 3 X a X 2 × b = 3 × 2 × a b = 6 a b. As the product is the same in whatever order the factors are arranged, we have simply changed their order and united in one product the numerical coefficients. Hence, when both factors are monomials, RULE. Annex the product of the literal factors to the produc of their coefficients, remembering that like signs give + and unlike, 2. Multiply a3 by a2. OPERATION. a3× a2= (a XaXa) X (a Xa) = a×a×a×a×a = a As the exponent of a quantity shows how many times it is taken as a factor, a3 — a × a × a; and a2: axa; and a3 X a2 = ax a×a×a×a, and this is equal to a3. (Art. 24.) Hence, Powers of the same quantity are multiplied together by adding their exponents. 17. Multiply together 444 x y, 3x2 y3, and 2 z. Ans. 18. Multiply 4 a2 b2 c d by 4 a b c2 d2. 19. Multiply 5 xm by 6 x2. 2664 x3 y* z. Ans. 30 m +n 5 a2bc, and 4 a b2. 22. Multiply 4 (x + y) by 3 (x + y). Ans. 75 bxay. Ans. 12 (x + y)2. NOTE. Any number of terms enclosed in a parenthesis may be treated as a monomial. 23. Multiply 12 (ab) by 4 (a2 b2). 24. Multiply (ax) by (ax)2. (a2 — Ans. 48 (a2 b2)2. 25. Multiply 4 (a + b)m by 2 (a + b)". Ans. 8 (a+b)m+n. 26. Multiply a3 (x + 2)2 by a b2 (x + z). |