6. 2 a 3(a - 1). 10. 1 [5(a - b) + 6(a + b)]. 1-[5(a b)+6(a 11. 5x-3(x-2y)-7[5x-3(x-3y)]. 12. a + a(1+ a2) — a[1 — a(1 − a)]. 21. Multiply (x2 + 1)1 − 5 α (x2 + 1)2 + 3 ab by — 2 a2b2 (x2 + 1)3. Simplify the result of substituting a + b − c for x, and a − b + c for y, in the following expressions: Find the values of the results of Exx. 22-24, 25. When a = — – 2, b = 3, c = — 4. Multiply 5 x 3 xn−3y2 + 4xn−5y4 + yn−4 by 27. x3. Write the squares, the cubes, and the nth powers of: 24. 7 abx+2 bcy. 30. - 6xnym. 34. -3 am+n-ly3. The Distributive Law when the Multiplier is a Multinomial. 12. Ex. (2+3) (7 − 5) = (2 + 3)7 − (2 + 3) 5 In general =2x7+3×7-2x5-3 x 5. (a + b) (c + d − e) = ac + bc + ad + bd — ae — be, etc. For, (a+b)(c + d − e) = (a + b)c + (a + b)d − (a + b)e, by Art. 10, Similarly for any number of terms in either multiplier or multiplicand. Multiplication of Multinomials by Multinomials. 13. From the preceding article is derived the following principle for multiplying a multinomial by a multinomial: Multiply each term of the multiplicand by each term of the multiplier, and add algebraically the resulting products. Ex. 1. Multiply - 3a+26 by 2a-3b. 2b We have (-3a+2b)(2 a − 3 b) = (-3a) × 2a +26 × 2a -3ax (-3b)+2b × (−3b) The work may be arranged as follows: Write the multiplier under the multiplicand, the first partial product, i.e., the product of the multiplicand by the first term of the multiplier, under the multiplier, the second partial product under the first, and so on, placing like terms of the partial products in the same column. We have ·3a + 2b 2 a 3b -6a2 + 4 ab +9 ab-6 b2 — 6a2 + 13 ab — 6 b2 It is customary to multiply from left to right, instead of from right to left as in Arithmetic. Ex. 2. Multiply a3+x3-4a2x-2 ax2 by x-3 a. Ex. 3. Multiply a3-3 ab+3ab2 - b3 by a2 - 2 ab + b2. a — 5 a1b + 10 a3b2 10 a2b3 +5 ab1 — b5 In this work the literal parts of the first three terms of the second and third partial products were omitted, it being understood that the numerals remaining are the coefficients of the literal parts just above in the first and second partial products. Observe that in the last example the multiplicand and multiplier, and also the product, are homogeneous. Ex. 4. Multiply 2x+1-5x+7xm-1- 9xm-2 by x2m — x2-1. We have Multiply 2x3m+1-723m + 12x3m-1-16 23m−2+923m-8 1. x + 3 by x + 7. by 33. 34. by 28. x2 - xy + y2 + x + y + 1 by x + y − 1. 29. 33 a+b+2 a2b8-44 a3b2 by 23 a3 - 21a2b+ ab2. 4 ar by 2 ar−3 + 3 ar−4. an-sxs + an by a2xn−1 — 3 xn+1 — 2 ax”. 43. (1+x+xm) (3 − 2 x2 + xm) (5 xn−1 – 3 xm-1). 14. The converse of the Distributive Law evidently holds; that is, ab + ac ad = a(b+c-d), etc. E.g., ax + bx= (a + b) x, 2 ay - 3 by = (2 a 15. If the coefficients of the multiplicand and multiplier, arranged to a common letter of arrangement, be literal, it is frequently desirable to unite the terms of the product which are like in this letter of arrangement. Ex. Multiply x + a by x + b. We have x + a x + b x2 + ax bx+ab x2+ ax + bx+ab = x2 + (a + b) x + ab, by Art. 14. EXERCISES XII. Arrange the values of the following products to descending powers of That is, a product is 0 if one of its factors be zero. 17. The words is not equal to, does not have the same value as, etc., are frequently denoted by the symbol E.g., 72, read seven is not equal to 2. 18. It follows, conversely, from Art. 16: . If a product be 0, one or more of its factors is 0. 1. What is the value of 2 (a - b), when b = a ? 2. What is the value of (a + b)(c – d), when c = d? 3. What is the value of (b+c) (a + b − c), when c = a + b? 4. What is the value of (x2 — 9) (x1 — 7 x3 + 2 x − 9), when x = 3? When x=- - 3? |