ORAL EXERCISES State the sum in each of the following exercises. 1. 2 x and – 3 x. . 2. 2 r and -5 r. 3. —3 r and 7 r. 4. –4 x, –5 x, and – 6 x. -3 y’, 2 y’, and — 7 y. 6. 6 A, 7 A, and – A. 7. –3x, – 7 x, and 10 x. 8. 4 ab, 7 ab, and – 2 ab. 9. rs, 3 rs?, and – 10 rs?. 10. -S, - 10 S, -4 S, and 21 S. 11. 6 r2x, 4 r2x, – 7 r2x, and – 3 r2x. 12. aby, – aby, 4 aby, and – 4 aby. 13. –4 ab 14. —rs 15. 3 aby 16. x Spa 7 ab 10 rs - 2 aby - 4 x3r -8 ab - 3 rs - 7 aby –7x2r 4 aby -3 xör 2 ab 2 rs WRITTEN EXERCISES Simplify the following expressions by uniting terms. 1. 6 x— 2 x+4 x– 7 x+10 x. [Hint. This is the same as 6 x+(-2 x)+4 x+(-7 x)+10 x.] 2.–26+3 6+7b-b7b. 3. 6 ar– 7 ar+15 ar-2 ar-ar. 4. 11 x4y— 2 x4y+8 x4y—3 x4y+2 x^y. 5. 10 xz+11 x2 – 4 xz+2 xz+3 x2–9 xz. 6. 8 A+10 A-3 A-6 A+4 A-12 A. 7. – aétia– a?. 8. į xy + } xy-{ xy. 9. 73 - p3- To r3. 10. ic+*c- c-c. 11. 5.7 2–2.3z+8.32. 12. 6.17 2+2.13 2–3.04 z. 13. – 1.5 x2z+6.5 x22— x22. 14. -5 A+2.5 A -1.7 A. 15. 2.3 x2–1.2 x2+.4 x2–2.1 x2. 16. 15(x+y) – 2(x+y)+6(x+y)-8(x+y). 30. Adding Unlike Terms. When terms are unlike, that is, when they do not all have a common factor, we can no longer express their sum in one term. In such cases we can only indicate the addition. Thus, the sum of 3 x and 4 y must be written out in the form 3 x+4y. In the same way, the sum of 5a-by and 2 rst is 5 ażby+2 rst. WRITTEN EXERCISES Express each of the following expressions in as few terms as possible. 1. 3 x+2y-7 X – 3 x+4 y. SOLUTION. Adding the x terms alone gives 32–73–3x=-7x. Adding the y terms alone gives 2y+4y=6y. Hence the result is – 7x+6y. Ans. 2. 2 a-36-4 a-3 b. 3. 4x+6–2x+7+9 x– 15. 4. 23 r- 15+9r+6-17 r–2. 5. 8 x’y– xy2 +7 xy2 — x’y — 4 xy?. 6. a-2b+2 c-2a+3 b-4c. 7. 2 a+2 6+2 c+2 d-a-3b-c-3 d. 8. mno+2 m?n?02 — 3 mno - 4 m?n?o2+g. 9. 1.2 a-2.4 b+2.3 a +1.5 b. 10. 3.26 x2 – 2.5 y2 – 1.75 x2. For further exercises on this topic, see Appendix, p. 294. 31. Polynomials. An expression which contains more than one term is called a polynomial. For example, 3 x+y, 3x+2y-42, and 6 x-y+2 2-m are polynomials. When a polynomial contains only two terms it is called a binomial. For example, 3 x+y, x+y, 4a-6, 6p+4q, are binomials. When a polynomial has only three terms it is called a trinomial. For example, 3x+2y-42, m+n-p, 48-6h+2i, are trinomials. 32. Addition of Polynomials. Polynomials are added by uniting terms that are alike. The process is similar to the adding of denominate numbers in arithmetic. For example, adding 6 yd.+1 ft.+ 3 in. and 2 yd.+1 ft.+ 8 in. gives 8 yd.+2 ft.+11 in. Ans. In the same way, adding 2 at 7 6+3 c and 6 at 3 6+2 c gives 8 a+10 6+5 c. Ans. These illustrations show that the rule for adding polynomials is as follows: RULE FOR THE ADDITION OF POLYNOMIALS. Write like terms in the same column, find the sum of the terms in each column separately, then connect the sums thus obtained by the proper signs. 33. Arrangement of Terms in a Polynomial. A polynomial is said to be arranged in the descending powers of some letter when the exponents of that letter decrease as we read the polynomial from left to right, as, for example, in X4+3 23+2 x2+7 x+9. A polynomial is said to be arranged in the ascending powers of some letter when the exponents of that letter increase as we read the polynomial from left to right, as, for example, in 9+7 x+2 x2 +3 x3 + x4. Before adding polynomials which contain several powers of the same letter it is best to arrange all terms according to the descending (or ascending) powers of that letter. Thus, the expressions 7 x—4 x2–9+x3 and -5 22—4 x+3 23+7 may be added as follows. Note that the terms are first arranged according to the descending powers of x: 23—4 x2+7 x-9 3 23—5 x2—4 x+7 4 23–9 x2+3 x-2. Ans. 34. The Checking of Addition. To check, or test, the work of addition we use special values of the letters and see if the result is correct for such values. This is illustrated in the following example. EXAMPLE. Add the expressions 3 a+4 6+2 c, 5 a+36-2 c, and 7 a-96-50 and check your answer when a=1, b=2, and c=3. SOLUTION. Adding, as in § 32, we have 3 a+4 6+2 c CHECK. When a=1, b=2, c=3 the value of 3 a+4 6+2 c is 3x1+4x2+2x3, which reduces to 17. Likewise, 5 a+3 6–2 c then becomes equal to 5, while 7 a-96–5 c becomes equal to – 26. The sum of the three is, therefore, 17+5–26, which reduces to – 4. But, the answer obtained above, namely 15 a-2 6-5 c, when likewise considered for a=1, b=2, c=3, becomes 15X1-2x2–5x3, or 15–4–15, and this also reduces to –4. Since the two results are the same (that is, each is – 4) the work checks. WRITTEN EXERCISES Add the following 1. 2a-36 3 a+86 (Check when a=1, b=1.) 2. –4 x+3 y 7 x–8 y (Check when x=2, y=1.) 3. 3r+2 stot 8r-58-9t (Check when r=1, s=2, t=3.) 4. –5 H+ I +10 K 7 H -9 1 + 8 K 3 m+ 7 n+ 8 p - 5 m + 4n-10 p 9 m-11 n+.5 p (Check when m=2, n=2, p=2.) 11. 3r+48+2 t and 6 r-3 s+5 t. 12. 69+5r+2, 9-5 r–2, and — 79+2r+2 z. 13. 8 x2 +4 x+7, 2 x2 – 3 x–5, and x2 — x-1. 14. a4+5 a2+3 a, 2 a3+6 a? — a, and a3+2 a+1. [Hint. The answer should contain five terms.) 15. 1-2r+3 r2, 2+3 r+4 ro, and -1+5r-3 r2. 16. x+4 x3— 2–5 x2, 6–2x+3 x2 — x3, and 2+x2+3x3 — X. [Hint. See § 33.] |