52. Division of Polynomials. Dividend. Divisor. -8a-8 o Remainder. We first arrange the dividend and divisor with reference to a (8 44), placing the divisor on the right of the dividend. Divide the first term of the dividend by the first term of the divisor. The result will be the first term of the quotient, which, for convenience, we place under the divisor. The product of the divisor by this term (6 a2 + 6 a), being subtracted from the dividend, leaves a new dividend, which may be treated in the same way as the original one; and so on to the end of the operation. Since all similar cases may be treated in the same way, we have, for the division of polynomials, the following rule: Arrange the dividend and the divisor with reference to the same letter. Divide the first term of the dividend by the first term of the divisor, for the first term of the quotient. Multiply the divisor by this term of the quotient, and subtract the product from the dividend. Divide the first term of the remainder by the first term of the divisor, for the second term of the quotient. Multiply the divisor by this term, and subtract the product from the first renainder, and so on. Continue the operation until a remainder is found equal to 0, or one whose first term is not divisible by that of the divisor. NOTES. — 1. When a remainder is found equal to 0, the division is exact. 2. When a remainder is found whose first term is not divisible by the first term of the divisor, the exact division is impossible. In that case, write the last remainder after the quotient found, placing the divisor under it, in the form of a fraction. (2) Let it be required to divide 51a’b? + 10 a* - 48ab - 1564 + 4 ab3 by 4 ab - 5 a? + 36?.' NOTE. — First arrange the dividend and divisor with reference to a. Dividend. Divisor. 10 a* - 48 ab + 51 a?b% + 4 abs – 1564 – 5a + 4 ab +363 + 10 a* - 8ab — 6 a’b? - 2a + 8 ab - 56% - 40a3b + 57 ab+ 4ab3 – 1564 Quotient. - 40a3b + 32a2b3 + 24 abs 25 a?b? — 20 ab — 1584 + xạy + xyz (3) 2* + x®y + x'y + xy’ – 2y x+y ** + x®y x +y + xy + xy - 24 NOTE. — Here the division is not exact, and the quotient is fractional. 1+ a 1-a +2a + 2a? + 2a’ – 2a + 2a NOTE. — In this example the operation does not terminate: it may be continued to any extent. 1. a2 + 2ax + 22 by a + x. Ans. a + x. 2. a' — 3a'y + 3 ay’ — y by a- y. Ans. a' — 2 ay + y. 3. 24 a’b – 12acb? — 6 ab by – 6 ab. Ans. – 4a + 2a2cb + 1. 4. 62* — 96 by 3x – 6. Ans. 2x3 + 4x2 + 8x + 16. 5. a — 5 a*x + 10 a®x? — 10 aʻx + 5 ax* — 205 by a' — 2 ax + x®. Ans. a* — 3 aʼx + 3 ax? — **. 6. 48.2* — 76 ax? — 64 aʼx + 105a3 by 2x – 30. Ans. 24 x? — 2ax — 35 a’. 7. — 3y^2 +3 y*2* — 206 by yo — 3 yox + 3yx? — 29. Ans. yö + 3y*x + 3yx? + 2 8. 64 a 86 — 25 a’68 by 8ab3 + 5 ab. Ans. 8a2b3 – 5 ab. 9. 6a +23 a’b + 22 ab: +563 by 3a’ + 4 ab +?. Ans. 2a + 56. 10. 6ax® + 6 ax+y + 42a*x by ax + 5 ax. Ans. 200 + xy + 7 ax. 11. – 15 a* +37a2bd — 29 aʼcf – 206'd' +44bcdf - 8c+f? by 3a: — 5 bd + cf. Ans. – 5a? +4bd — 8cf. 12. x + x*y* + y4 by 2 - xy + y? Ans. 2 + xy + y'. 13. 2* — yo by x - y. Ans. 2 + xạy + xy? +yo. 14. 3a' – 8a+b? + 3 a'c +56 - 36°cby a' – 6?. Ans. 3a- 56° +3c. 15. 6.2.– 5x*y* - 6x*y+ 6.x*y* + 15x*y* – 9.2*y* + 10x*ys + 15yo by 320 + 2x*y* + 3y. Ans. 220 – 32*+ 5 y. CHAPTER III. FACTORING, GREATEST COMMON DIVISOR, AND LEAST COMMON MULTIPLE. USEFUL FORMULAS. 53. A formula is an algebraic expression of a general rule or principle. Formulas serve to shorten algebraic operations, and are also of much use in the operation of factoring. When translated into common language, they give rise to practical rules. The verification of the following formulas affords additional exercises in multiplication and division. 54. Formula 1. – To form the square of a +6, we have (a+b)=(a + b)(a+b)= a + 2ab + b2 = a + b3 + 2 ab; that is, The square of the sum of any two quantities is equal to the sum of their squares, plus twice their product. (1) Find the square of 2a+36. We have from the rule, (2a +36)2 = 4 a2 +962 +12 ab. (2) Find the square of 5 ab + 3ac. Ans. 25 a’b? + 9 a'c + 30 a'bc. (3) Find the square of 5a' +8a2b. Ans. 25 a* + 64 a'b? + 80 a*b. (4) Find the square of 6 ax +9a2x. Ans. 36 aʼx2 + 81 a*x+ + 108 a*. |