represent the same value. The following rules are agreed upon: I. A succession of multiplications and divisions shall be performed in the order in which they occur. Thus 68÷4.3 = 484.312 .336. II. A succession of additions and subtractions shall be performed in the order in which they occur. III. A succession of multiplications, divisions, additions, and subtractions shall be performed in accordance with I and II; the multiplications and divisions being performed before any additions and subtractions. Thus, 20 +6.7 43.520 +42 4-15= 43; 17 +15 ÷ 3 8 2 ÷ 4 = 175-16 ÷ 4 = 1754 18. 6. If there are parentheses, apply the preceding rules to the expressions within the parentheses first; then to the resulting expression as a whole. The forms (), [], { } go by the general name of “ parentheses" but they are designated by special names when it is necessary to distinguish one from the other. Thus, [] is called a "bracket," { } a "brace," but () is always called a "parenthesis." MULTIPLICATION 8. To multiply one term by another: Multiply the numerical coefficients; annex the letters, giving to each letter in the product an exponent equal to the sum of its exponents in the two factors. Law of signs: Like signs give plus, unlike signs give minus. The degree of a term is the sum of the exponents of the literal factors. Thus 4 xy2z8 is of the sixth degree. An expression is homogeneous, if all its terms are of the same degree. In x+2x22 — 5 xy3 every term is of the fourth degree; the expression is homogeneous. To multiply a polynomial by a polynomial: Arrange the terms in both polynomials according to the ascending or descending powers of some letter; multiply each term of the multiplicand by each term of the multiplier and add the partial products. Multiplication may be performed by detached coefficients. For example, (2 x4 - 7 x3- 5 x2+6x-3) (6 x2-4x+5). The product is 12 26 - 50x58x +21 x3- 67 x2+42x-15. If any powers of x are lacking in either polynomial, the terms in question must be represented by zeros. Thus, in multiplying x3 + 2 x − 4 by x2 + x2 + 1, we must write 1+0+2-4 and 1 + 0 + 1 + 0 + 1. Check an example in multiplication by substituting numerals for the letters. DIVISION 9. To divide one term by another: Divide the numerical coefficients; annex the letters, giving to each letter in the quotient an exponent equal to its exponent in the dividend minus its exponent in the divisor. Observe the law of signs: Like signs give plus, unlike signs give minus. To divide one polynomial by another: I. Arrange the terms. II. Divide the first term of the dividend by the first term of the divisor and obtain the first term in the quotient. III. Multiply the divisor by this term in the quotient. IV. Subtract the product from the dividend. V. Treat the remainder as a new dividend and proceed as before. VI. Keep each new dividend arranged in the same order as the first dividend. Divide 14 x4 - 27 ax3 + 21 a2x2 - 32 a1 + 12 a3x by 2 x2 - 3 ax + 4 a2. By detached coefficients this division may be performed as follows: Division may be checked by substituting numerals for the letters, care being taken to avoid a zero divisor. ORAL EXERCISES 10. 1. In what other way may examples in multiplication and division be checked? 2. If there is a remainder in a division, what must be done with it in checking? 3. If the multiplicand and the multiplier are arranged according to the descending powers of some letter, how will the product be arranged? 4. If the multiplicand and the multiplier are homogeneous, what will be true of the product? Of what degree will it be? 5. 12 xa+4+20 xa +3 — 107 x +2 +37+1+33 x by 6 x3 — 11 . 6. m3 + n3 + p3 – 3 mnp by m+n+p. 7. a3 — b3 + c3 +3 abc by a2 + b2 + c2 + ab ac + bc. 8. y5 — 5 y1 + 2 § y3 — § 1 y2 + § y − 1 by y — 3. 9. c − c3 + 1⁄2 c1+254c3-13 c2 + 13 c - by c- c2 + } c −2. 1 1 PARENTHESES 12. I. A parenthesis preceded by a + sign may be removed. without changing the signs of the terms within the parenthesis. II. A parenthesis preceded by a - sign may be removed, provided that the sign of each term within the parenthesis be changed. 3. 8x [2x+(3 x − y) +7] + [5x-2y+3]. 4.7 m + [2 n − {4 m - (3 m − 5 n) — 8 n} + 9 m]. 5. 10 c4d (9 c-5 d)} - {(2 cd) - (4c+7d)}. 6. (x+2)(x −3) — 5(x2 − 3 x + 2) − 10 (x + 3). 7. {5 a2 - 3[−5 + (a + 2) (a − 5) — 7]} — a(a − 1)2. 8. (y-3)3 — y2 (y − 3)2 + ( − y2 + 7 y + 8). 9. (a + b)(a − b) — (a − b)2 + (a + b)2. 10. 2-4 (7 −3) + (-8-2)-5(9-3+5)+11. 11. m—{— [—(1—m)—1]—m} — { m — (5 — 4 m) — (4+m)}. 14. Terms may be inclosed in a parenthesis by reversing the rules for removing parentheses. When terms are inclosed in a parenthesis preceded by a + sign, the signs of the terms are not changed. When terms are inclosed in a parenthesis preceded by a sign, the signs of all the terms are changed. WRITTEN EXERCISES 15. Collect the coefficients of a in a parenthesis preceded by a sign, and the coefficients of y in a parenthesis pre ceded by a 1. ax sign: 2. mx + nx + my — ny. 3. abx + cdy-cdx-aby. |