37. In the following examples the pupil is required to translate the algebraic symbols into common language. Ans. The quotient arising from dividing the sum of a and x by b, increased by the quotient of x divided by c, is equal to the quotient of m divided by the sum of a and b. Ex. 2. 7a2+(b-c)x(d+e)=x+y. How should the preceding example be read when the first parenthesis is omitted? Computation of Numerical Values. 38. The numerical value of an algebraic expression is the result obtained when we assign particular values to all the letters, and perform the operations indicated. Suppose the expression is 2a2b. If we make a=2 and b=3, the value of this expression will be 2x2x2x3=24. If we make a=4 and b=3, the value of the same expression will be 2x4x4x3-96. The numerical value of a polynomial is not affected by changing the order of the terms, provided we preserve their respective signs. The expressions a2+2ab+b2, a2+b2+2ab, and b2+2ab+a2, have all the same numerical value. Find the numerical values of the following expressions, in which a 6, b=5, c=4, m=8, and n=2. Ex. 1. a2+3ab-c2. Ex. 2. a2x(a+b)—2abc. Ans. 36+90-16=110. CHAPTER II. ADDITION. 39. Addition, in Algebra, is the connecting of quantities together by means of their proper signs, and incorporating such as can be united into one sum. When the Quantities are similar and have the same Signs. 40. The sum of 3a, 4a, and 5a, is obviously 12a. That is, 3a+4a+5a=12a. So, also, -3a, -4a, and -5a, make -12a; for the minus sign before each of the terms shows that they are to be subtracted, not from each other, but from some quantity which is not here expressed; and if 3a, 4a, and 5a are to be subtracted successively from the same quantity, it is the same as subtracting at once 12a. Hence we deduce the following RULE. Add the coefficients of the several quantities together, and to their sum annex the common letter or letters, prefixing the common sign. The pupil must continually bear in mind the remark of Art. 26, that, when no sign is prefixed to a quantity, plus is always to be understood. When the Quantities are similar, but have different Signs. 41. The expression 7a-4a denotes that 4a is to be subtracted from 7a, and the result is obviously 3a. That is, 7a-4a=3a. The expression 5a−2a+3a—a denotes that we are to subtract 2a from 5a, add 3a to the remainder, and then subtract a from the last sum, the result of which operation is 5a. That is, 5a-2a+3a-a=5a. It is generally most convenient to take the sum of the positive quantities, which in the preceding case is 8a; then take the sum of the negative quantities, which in this case is 3a; and we have 8a-3a, or 5a, the same result as before. Hence we deduce the following RULE. Add all the positive coefficients together, and also all those that are negative; subtract the least of these results from the greater; to the difference annex the common letter or letters, and prefix the sign of the greater sum. When some of the Quantities are dissimilar. 42. Dissimilar terms can not be united into one term by addition. Thus 2a and 36 neither make 5a nor 5b. Their sum can therefore only be indicated by connecting them by their proper signs, thus, 2a+3b. In adding together polynomials which contain several groups of similar quantities, it is most convenient to write them in such a manner that each group of similar quantities may occupy a column by itself. Hence we deduce the following RULE. Write the quantities to be added so that the similar terms may be arranged in the same column. Add up each column separately, and connect the several results by their proper signs. EXAMPLES. 1. Add 2a+3b+4c, a+2b+5c, 3a-b+2c, and −a+4b-6c. Ans. 5a+8b+5c. 2. Add 2xy-2x2+y2, 3x2+xy+4y2, x2—xy+3y2, and 4x2 -2y2-3xy. Ans. 6x2―xy+6y2. 3. Add 5a2x2-2xy, 3ax-4xy, 7xy-4ax, a2x2+5xy, and 2ax-3xy. Ans. 6a2x2+3xy+ax. 4. Add 2a2-3ac+3b-cd, 4a2-ac+2cd-b, 3a2+2ac-4b +3cd, and a2-2ac+5cd-2b. Ans. 10a2-4ac-4b+9cd. 5. Add 7m+3n-14x, 3a+9n-11m, 5x-4m+8n, and 11n-2b-m. Ans. 3a-2b-9m+31n-9x. 6. Add 2a2x+3ax2+x2, 2ax—3a2-4x2, −2x2+3a2x-5ax2, and 3a2-2a2x+2αx2. Ans. 3a2x-5x2+2ax. . 7. Add 2a2b2-3ax+5m2y, 2m2y+3a2b2—2ax, 4ax-3m2y -4a2b2, and ax+3a2b2—4m2y. Ans. 4a2l2. 8. Add 7ab3-12ax2, 13ab3+ax, 3ax2-8ab3, and -12ab3 +9ax2-5. 9. Add 4x2+2ax+1, 3ax-2x2+5, 3x2-6x+4, and 5x2 +ax-1. 10. Add 2a3x2-3a2x3 — a2x-ax2+2ax, 4a2x3 — 3a2x+-4a3x2 +2ax2+3ax, and 3a3x2-ax2-a2x3+4a2x+ax. 11. Add 14a3x — 7a2b2 + 3a2, 5a2b2c2 + 3a2b2+2a2, 2a2b2c2 -5a3x-a2, and 4a2b2-9a3x-4a2. 12. Add ax1-bx3+cx2-7, 2bx3 +3cx2-4x+1, 3αx1-4bx3 -2cx2+3, and 2ax1+3bx3-2cx2+3. 43. It must be observed that the term addition is used in a more general sense in Algebra than in Arithmetic. In Arithmetic, where all quantities are regarded as positive, addition implies augmentation. The sum of two quantities will therefore be numerically greater than either quantity. Thus the sum of 7 and 5 is 12, which is numerically greater than either 5 or 7. But in Algebra, the quantities to be added may be either positive or negative; and by the sum of two quantities we understand their aggregate, taken with reference to their signs. |