CHAPTER II. FUNDAMENTAL OPERATIONS. I. ADDITION. Definitions. 16. Addition is the operation of finding the simplest expression for the aggregate of two or more quantities. This expression is called their sum. Explanation. 17. If the quantities are similar, the addition may be performed; if they are not similar, the operation can only be indicated. Thus, the sum of 7a2b and 4a2b, is 11a2b, in the same way that the sum of 7 pounds and 4 pounds, is 11 pounds. The sum of 3a2c and 4b2, cannot be expressed by a single term, any more than the sum of 3 pounds and 4 feet; the sum may, however, be indicated by writing the quantities one after another, with their proper signs; thus, 3ac2 + 4b2. Addition of Similar Terms. 18. To deduce a rule for adding similar terms, let us take the following examples: In the first example the unit a2bc is taken positively 5 times, 3 times, 1 time, and 7 times, that is, it is taken positively 16 times; hence, in this case, the sum of the monomials is +16a2bc. In the second example the same unit is taken negatively 8 times, 1 time, 3 times, and 2 times, that is, it is taken negatively 14 times; hence, the sum of the monomials, in this case, is 14a2bc. In the third example the unit a2bc is taken positively 11 times, and negatively times, that is, it is taken positively 4 times more than it is taken negatively; hence, the sum, in this case, is + 4a2bc. In the fourth example, the unit is taken negatively 10 times, and positively 8 times, that is, it is taken negatively 2 times more than it is taken positively; hence, the sum, in this case, is - 2a2bc. Since we may treat all similar cases in the same manner, we have the following rule for adding similar terms: RULE. Add the coefficients of the positive and negative terms separately; subtract the less sum from the greater, prefixing the sign of the greater; to the result annex the common literal part. EXAMPLES. 1. Find the sum of 3ays, -5ay3, -2ays, and 7ays. Ans. 3ays. 2. Find the sum of 4cz, - cz4, 3cz4, and 14cz4. 3. Find the sum of 8bc2, -4bc2, -11bc2, and -2bc2. 19. The sum of two or more polynomials may be found by first writing them one after another with their proper signs, and then reducing similar terms to their simplest form by the preceding rule. In practice it is found more convenient to write the polynomials. so that each group of similar terms may be found in a single column; hence, we have the following rule for the addition of polynomials: RULE. I. Write the quantities to be added so that similar terms shall fall in the same column. II. Add each column of similar terms separately, and to the result annex the dissimilar terms with their proper signs. Find the sums of the following groups of polynomials: 1. a+b+c, a + b―c, a-b+c, and a + b + c. Ans. 2a 2b + 2c. 2. 2ax + 3by, 3ax + 2by, ax + by, and Sax + by. Ans. 20ax + 13by. 3. 2a2-17ab + 362, 5a2+12ab5b2, and 362 + 6ab + 3a2. 6ab + 12a2- 962, Ans. 22a2+ Yab 862. 5. 2x+3y-42—10, 8y—4x+7z+8, 11z+5x−10y—2, and 16+10x + 12y + 14z. 6. 3x3 + 2y3 + 23 + 8xyz, Ans. 13x+13y+28z+12. y3 + 2x3-3x3- 4xyz, Ans. 923+2y3 + 3xyz. 23 + 3x3 — 2y3 — 2xyz, and zyx + x3 + y3 + 23. 23 7. x + 3x3y + x2z — 2xv, 30x4 - 29x2 + 18xv-17x3y, 22x3y-15x-32x2+16xv, and 17x2z-12x+6x3y-11xv. 9. ax + 2bx+4by — 3ay, 2ax + bx +2ay - by, and 4ax + 3by. Ans. Tax +3bx+6by-ay. 10. px+qy+rz-c, 2px - 2qy + 2c, 3qy - px +4c, 13. a-x+4y — 3z + w, z — w — y x + y + z. За - 2x2 + ax. |