Subtract 9. 3x-5y-7z from 2x + 3y - 4z. 10. -4x-2y+11z from -x+2y-13z. 13. m-2n-p from m+2n. 14. 2p-3q-r from 2q-4r. 15. ab-2cd-ac from ab - 3cd+2ac. 16. 3ab+6cd3ac-5bd from 3ab+5cd-4ac-6bd. 17. − xy+yz -zx from 2xy + zx. 18. -2pq-3qr+4rs from qr – 4rs. 19. - mn+11np - 3pm from -11np. 20. x2y-2xy2+3xyz from 2x2y + 3xy2 − xyz. From 21. x3-3x2+x take - 23 +3x2 − x. 22. 23. 2x3- x2 take x3- x2-x. a3+13-3abc take b3-2abc. 24. -8+6bc+b2c2 take 4-3bc5b2c2. 27. -4+x2y-xyz take 3-2x2y +11xyz. 31. 2+3x-7x2 take 3x2-3x-2. 38. 1-x+x5-24-23 take x − 1 + x − x2. 39. - Smn2+15m2n+n3 take m3 - n3+8mn2 - 7m2n. 40. 1-p3 take 2p3-3pq2 -2q3. 33. The following exercise contains miscellaneous examples on the foregoing rules. MISCELLANEOUS EXAMPLES I. 1. When x = 2, y = 3, z = 4, find the value of the sum of 5x2, -3xy, z2. Also find the value of 3*+3x. 2. Add together 3ab+bc- ca, ab+ca, ab-2bc+5ca. the sum take 5ca + bc - ab. From 3. Subtract the sum of x-y+3z and -2y-2% from the sum of 2x-5y-3z and −3x+y+4z. 4. Simplify (1) 36 - 2b2 − (2b – 362). (2) 3a-2b-(2b+ a) − (a − 5b). 5. Subtract 8c2+8c-2 from c3 – 1. 6. When x = 3, a = 2, y = 4, % = 0, find the value of 7. Add together 3a-7a+5 and 2a3+5a-3, and diminish the result by 3a2+2. 8. Subtract 262-2 from -2b+6, and increase the result by 3b-7. 9. Find the sum of 3x2-4x+8, 2x-3-x2, and 2x2-2, and subtract the result from 6x2+3. 10. What expression must be added to 5a2 - 3a +12 to produce 9a2-7? 11. Find the sum of 2x, -x3, 3x2, 2, −5x, −4, 3×3, -5x2, 8; arrange the result in ascending powers of x. 12. From what expression must the sum of 5a2-2, 3a+a2, and 7- 2a be subtracted to produce 3a2 + a -- 5 ? 13. When = 6, find the numerical value of the sum of 1−x + x2, 2x2-1, and x − x2. 14. Find the value of 6ax + (2by – cz) – (2ax – 3by +4cz) − (cz+ax), when a 0, b = 1, c = 2, x = S, y = 3, z = 4. = 15. Subtract the sum of 23 - 3x2, 2x2 - 7x, 8x-2, 5-3x3, 2x3-7 from x3 + x2+x+1. 16. What expression must be taken from the sum of p − 3p3, 2p+8, 2p2, 2p3-3p, in order to produce 4p-3 ? 17. What is the result when - 3x3 + 2x2 - 11x+5 is subtracted from zero. 18. By how much does b+c exceed b-c? 19. Find the algebraic sum of three times the square of x, twice the cube of x, ·x3+x- 2x2, and x23 − x − x2 + 1. 20. Take p2-q2 from 3pq-4q2, and add the remainder to the sum of 4pq - p2 - 3q2 and 2p2+6q2. 21. Subtract 3b3+2b2-8 from zero, and add the result to b4-263+3b. 22. By how much does the sum of 2m3 - 2m3 +5, 3m3 +4m2 +5m+3, fall short of m3 +2m - 1, m2 - 3m, 11m3 – Sm2+3m ? 23. Find the sum of 8x-4x3y2, 7x1y — xy1, 3x3y2+2x2y3 + 5xy1, y5 - 4xy1+x3y2, x5 −y3+x3y2+xy* - x2y3 +3x1y, and arrange the result in descending powers of 2. 24. To what expression must 3x-4x+7x2+4 be added so as to make zero? Give the answer in ascending powers of x. 25. Subtract 7x2-3x-6 from unity, and x - 5x2 from zero, and add the results. 27. Find the sum of a, -3a2, 4a, -5a, 7, -18a, 4a2, -6, and arrange the result in descending powers of a. - 28. Add together 4+3x+x3, x2- x2-11, x3- 2x2+7, and subtract 2a+a2 - 7 from the result. 29. If a=5x-3y+2, value of a+b-c. b= -2x+y-3, c=x- 5y+6, find the 30. If x=2a2 - 5a+3, y= −3a2+a+8, z=5a2 - 6a-5, find the value of x-(y + z). MULTIPLICATION. 34. MULTIPLICATION in its primary sense signifies repeated addition. Here the multiplier contains 5 units, and the number of times we take 3 is the same as the number of units in 5. Again ax ba taken b times =a+a+a+a+.. the number of terms being b. Also 3×5=5×3; and so long as a and b denote positive whole numbers it is easy to shew that Hence axb=bxa. abc=axbxc=(a× b)×c=b×a× c=bac Similarly we can show that the product of three positive integral quantities a, b, c is the same in whatever order the factors are written. Example. 2a × 3b = 2 × a × 3 × b = 2 × 3 ×a× b = 6ab. 35. When the quantities to be multiplied together are not positive whole numbers, the definition of multiplication has to be modified. For example to multiply 3 by 4, we perform on 3 that operation which when performed on unity gives; that is, we must divide 3 into 7 equal parts and take 4 of them. By taking multiplication in this sense, the statement ab=ba can be extended so as to include every case in which a and b stand for positive quantities. It follows as in the previous article that the product of a number of positive factors is the same in whatever order the factors are written. 36. Since, by definition, a3=aaa, and a=aaaaa; = that is, the index of a in the product is the sum of the indices of a in the factors of the product. Again, 5a2=5aa, and 7a3=7aaa ; -35α. .. 5a2 × 7a3=5×7 × aaaaα= When the expressions to be multiplied together contain powers of different letters, a similar method is used. Example. 5ab2 × 8a bx3 = 5aaabb × 8aabxxx Note. The beginner must be careful to observe that in this process of multiplication the indices of one letter cannot combine in any way with those of another. Thus the expression 40a5b3a admits of no further simplification. 37. Rule. To multiply two simple expressions together, multiply the coefficients together and prefix their product to the product of the different letters, giving to each letter an index equal to the sum of the indices that letter has in the separate factors. The rule may be extended to cases where more than two expressions are to be multiplied together. Example 1. Find the product of x2, x3, and x3. The product = x2 × x3 × x8 = x2+3 × 28 = x2+3+8 = x13. The product of three or more expressions is called the continued product. Example 2. Find the continued product of 52y3, 8y2z5, and 3xz. The product = 5x2y3 × 8y2z5 × 3xz1 = 120x3y5z9. 38. By definition, (a+b) m=m+m+m+ ... taken a+b times =(m+m+m+... taken a times), (m+m+m+... taken b times) =am+bm. together with Also diminished by Similarly (a−b+c) m=am− bm+cm. ... (a−b) m=m+m+m+. taken ab times =am- - bm. ... Thus it appears that the product of a compound expression by a single factor is the algebraic sum of the partial products of each term of the compound expression by that factor. Examples. 3(2a+3b - 4c) = 6a+9b-12c. (4x2 – 7y − S≈3) × 3xy2 = 12x3y2 – 21xy3 – 24xy2z3. |