CHAPTER IV. ADDITION AND SUBTRACTION. Integral Compound Expressions. 90. If an algebraic expression contains no letter in the denominator of any of its terms, it is called an integral expression. Thus, x3 + 7 cx2 — c3 — 5 c2x, † ax integral expressions. -bcy, are An integral expression may have for some values of the letters a fractional value, and a fractional expression an integral value. If, for instance, a stands for and b for , the integral expression 2 a 5 b stands for — § = = 5 a ; and the fractional expression stands for 15 5. Integral and fractional expressions, therefore, are so named on account of the form of the expressions, and with no reference whatever to the numerical value of the expressions when definite numbers are put in place of the letters. Addition of Integral Compound Expressions. 91. The addition of two compound algebraic expressions can be represented by connecting the second expression with the first by the sign +. If there are no like terms in the two expressions, the operation is algebraically complete when the two expressions are thus connected (§ 11, Note). If, for example, it is required to add m + n − p to a+b+c, the result will be a + b + c + (m + n − p); or, removing the parenthesis (§ 39), a + b + c + m + n − p. 92. If, however, there are like terms in the expressions, every set of like terms can be replaced by a single term with a coefficient equal to the algebraic sum of the coefficients of the like terms. 1. Add 5 a2 + 4 a + 3 to 2 a2 — 3 a — 4. This process is more conveniently represented by arranging the terms in columns, so that like terms shall stand in the same column, as follows: The coefficient of a2 in the result will be 5 + 2, or 7; the coefficient of a will be - 3+ 4, or 1; the last term will be — 4 + 3, or — 1. 2. Add 2 a3 – 3 a2b+4 ab2 + b3 ; a3 + 4 a2b — 7 ab2 — 2 b3 ; 3 a3+ab-3 ab2-463; and 2 a3 + 2 ab + 6 ab2 - 3 b3. — The coefficient of a3 in the result will be 2 + 1 + 3 + 2, or +8; the coefficient of a2b will be 3+4+1 +2, or + 4; the coefficient of ab2 will be 4 - 7 −3 +6, or 0, and, therefore, the term ab2 will not appear in the result (§ 21); and the coefficient of b3 will be 1 - 2 - 4 -3, or -8. + 5. ax2 bx-4; 3 ax2 - 2 bx +4; - 4 ax2 - 2 bx +5. 6. 5x+3y+2; 3x+2y+3z; x-3y-52. 7.3 ab 2 ax2 + 3 a2x + x3; 4 ab 6 a2x + 5 ax2; x3-aba2x-ax2; ax2+8 ab - 5 a2x. - 8. a 2 a3 + 3 a2 — a + 7; 2a 3a +2 a2 a +6; a-2 a3+2 a2 - 5. c2; − - 9. 3 a2ab+ ac3b2+4 bcc2; 4 bc+5 c2+2 ab; 5a2 abac + 5 bc; -4 a2 + b2 - 5 bc + 2 c2. 10. -3x+2x2-4x+7; 3x + 2 x3 + x2 - 5 x − 6; x1 4x+3x3-3x2+9x-2; 2x4x3 + x2 - x + 1. 11. 7 y3 — 3 xy2 — 4x2y + x3; − 5 x3 – 11 xу2 – 12 xz2 — y3; x2y-xx2- y2-5xy2; -4x2+ y2+6xy2+10x3. 12. a 2 a3 +3a2 - 3 a 2 ; 2; a3 + a2 + a − 3 a1 + 3; 4 a* +5 a3; 2 a2+3a-2; a2 - 2 a 3. 13. x+2xy2x2y - y3; 2 x3-3x2y - 4xy3 — 7 y3 ; 14. c − 3 c3 + 2 c2 4c+7; 2 c1 +3 c3 + 2 c2 + 5c+6; c3-4 ct 4 c2 - 5. 15. 3x2 - xy + xz - 3 y2 — x2; - 5x2xy - xz + 5 yz ; y2+3 yz + 3x2; 6x2-6y-6x+4xx; 4yz -5xz. 16. m3 - 3 m1n - 6 m3n2; m3n2 + m2 n3 — 5 m1n 3 no; – − 2 m3 + 4 m2n3 - 3 mn1 n5; 2 m2n3 - 3 mn1 + n3 ; — m3 + 2 mn1 + 2 n0 + 3 m1n. 17. 6y-1-2x2y; 5+2 xy2 - 4x2y; - x2y - 5+6xy2; 2 + xy-y3; x2y-2xy2-5 y3 +1. Subtraction of Compound Expressions. 93. The subtraction of one expression from another, if none of the terms are alike, can be represented only by connecting the subtrahend with the minuend by means of the sign If, for example, it is required to subtract a+b+c from m+n-p, the result will be represented by m + n − p − (a+b+c); or, removing the parenthesis (§ 40), If, however, some of the terms in the two expressions are alike, we can replace like terms by a single term : 1. Subtract as 2 a2 + 2 a 1 from 3 a3 - 2 a2 + a -2; the result may be expressed as follows: 3 a3-2 a2+a-2-(a-2 a2+2a-1); or, removing the parenthesis (§ 40), 3 a3-2 a2 + a 2-a3 +2 a2 2a+1 =3a3a3-2 a2 + 2 a2 + a-2a-2+1 (§ 38) 2 a3 a - 1. = This process is more easily performed by writing the subtrahend below the minuend, mentally changing the sign of each term in the subtrahend, and adding the two expressions. Thus, the above example may be written The coefficient of a3 will be 31, or 2; the coefficient of a2 will be2+2, or 0, and therefore the term containing a2 will not appear in the result; the coefficient of a will be 1 — 2, or 1; the last term will be 2+1, or — 1. |