28. Terms not differing excepting as to their numerical coefficients are like or similar terms. 5 ax and - 3 ax are similar terms. 3 ab and 14 mn are dissimilar terms. 29. The common expressions of algebra are frequently named in accordance with the number of terms composing them. The following names are generally used: A Monomial. An algebraic expression of one term. 4a, 5mn, - 3 xy, and 17 xyz are monomials. A Binomial. An algebraic expression of two terms. a + b, 3 m A Trinomial. An algebraic expression of three terms. 3 x 7 m + 8, 4 ab - 11 ac – 10 mny are trinomials. A Polynomial. Any expression having two or more terms. While the binomial and the trinomial both come under this head they occur so frequently that common practice gives each a distinct name. Expressions having four or more terms are ordinarily named polynomials. Oral Drill Read the following algebraic expressions: 1. a+3x-4 mn + cdn − 3 xy. 2. 2mx-3 acd+ (a + x) − (m+n). 4. 5ayz -2(2 m — n) + a(a−x)+3a(2x-3y). 5. − mnx+3 a(c− 2 d + 1) − (a− m+n)x+12(4 − x). 6. (a−x)(c+y)- (x+2) (y − 3) − (x+1)(x+2)(x+3). α x a_x_2 (c + d) _ 2 (m + n − p) − (c+1)o± 3. 7. + b y 3(c-d) c-y CHAPTER II ADDITION. PARENTHESES 30. Addition in algebra, as in arithmetic, is the process of combining two or more expressions into an equivalent expression or sum. The given expressions to be added are the addends. THE NUMBER PRINCIPLES OF ADDITION 31. The Law of Order. Algebraic numbers may be added in any order. 32. The Law of Grouping. The sum of three or more algebraic numbers is the same in whatever manner the numbers are grouped. In general: a+b+c=a+(b+c)=(a+b)+c=(a+c)+b. Numerical Illustration: 2+3+4=2+(3+4)=(2+3)+4=(2+4)+3. A rigid proof of these laws is not necessary at this point; but may be reserved for later work in elementary algebra. The law of order is frequently called the commutative law, and the law of grouping is called the associative law. ADDITION OF MONOMIALS The principles underlying the addition of the simplest forms of algebraic expressiors have already been developed, and they are readily applied in the more difficult forms of later work. (1) The sum of like quantities having the same sign, all + or 33. The coefficient of the sum of similar terms having like signs is the sum of the coefficients of the given terms with the common sign. . (2) The sum of like quantities having different signs, some + and some. 34. The coefficient of the sum of similar terms having unlike signs is the arithmetical difference between the sum of the + coefficients and the sum of the coefficients, with the sign of the greater. If the sum of three or more terms is required, we apply the law of grouping (Art. 32), and separately add the + and terms. the Thus, 5-8 +11-16 + 3 = 5 + 11+3816 = 19-24 - 5. Result. == Let the student apply this principle in the following: Add orally: 1. 2-76-9. 2. 8+3-12+7. 3.98-15-11. 4. 7-14-3+10. 5. 13-18+7-21. 6. 3a 11. 3a-5a+8a-11a-3a+a. 4a+6a-3 a. 12. -4 xy+3 xy-7 xy+xy-xy+8 xy. 13. 5 am - 8 am -24 am + 13 am am +6 am 11 am. 14. 15. 6m-7-4m+11-5m-17-m+5m+13. − 4 cx +8 cx − 3 cx + 2 cx − 3 cx + cx − 15 cx – 14 cx. -- ADDITION OF POLYNOMIALS The principles established for the addition of monomials apply directly to the addition of polynomials. Illustrations: 1. Find the sum of 5 a+7b-2c, 2a-3b+8c, and -3a+2b-10 c. It frequently happens that not all of the terms considered are found in each of the given expressions, in which case we arrange the work so that space will be given to such terms as an examination shows need for. 2. Add 4a+3b+3m, 2b+3c-d, 2a+3 d + 2 m −x, and 5 b-5 m-3 x. The coefficient of the m-term in the sum being 0, that term disappears. In general, to add polynomials: 35. Write the given expressions so that similar terms shall stand in the same columns. Add separately, in each column, the positive coefficients and the negative coefficients, and to the arithmetical difference of their sums prefix the proper sign. Collecting terms is another expression for adding two or more given expressions. 36. Checking results. The accuracy of a result may be checked by substituting convenient numerical values for each of the given letters. The substitution is made both in the given expressions and in the sum obtained, and the work may be considered accurate if both results agree. Illustration: To check the sum of 5a + 7b+ 2c and 2a - 3b - 5c, let a = 1, b = 2, and c = 1. |