CASE VII. A common polynomial factor. Any _composite_ number of terms. t^2 p + t^2 q - t^2 r - g^2 p - g^2 q + g^2 r = t^2 (p + q - r) - g^2 (p + q -r) = (p + q - r)(t^2 - g^2) = (p + q - r)(t + g)(t - g). CASE VIII. The Factor Theorem. Any number of terms. x^3 + 17x - 18 = (x - 1)(x^2 + x + 18). FACTORING Review the _Cases in Factoring_ (see Outline on preceding pages) and write out the prime factors of the following: 1. 8a^(13) + am^(12). 2. x^7 + y^7. 3. 4x^2 + 11x - 3. 4. m^2 + n^2 - (1 + 2mn). 5. -x^2 + 2x - 1 + x^4. 6. x^(16) - y^(16). (Five factors.) 7. (x + 1)^2 - 5x - 29. 8. x^4 + x^2 y^2 + y^4. 9. x^4 - 11x^2 + 1. 10. x^(2m) + 2 + 1/(x^(2m)). 11. x^(6m) + 13x^(3m) + 12. 12. 4a^2 b^2 - (a^2 + b^2 - c^2)^2. 13. (x^2 - x - 6)(x^2 - x - 20). 14. a^4 - 8a - a^3 + 8. 15. p^3 + 7p^2 + 14p + 8. 16. 18a^2 b + 60ab^2 + 50b^3. 17. x^3 - 7x + 6. 18. 24c^2 d^2 - 47cd - 75. 19. (a^2 - b^2)^2 - (a^2 - ab)^2. 20. a^2 x^3 - (8a^2)/(y^3) - x^3 + 8/(y^3). 21. gt - gk + gl^2 + xt - xk + xl^2. 22. (m - n)(2a^2 - 2ab) + (n - m)(2ab - 2b^2). 23. a^2 - x^2 - y^2 + b^2 + 2ab + 2xy. 24. (2c^2 + 3d^2)a + (2a^2 + 3c^2)d. 25. (n(n - 1))/(1 . 2) a^(n - 2) b^2 + (n(n - 1)(n - 2))/(1 . 2 . 3) a^(n - 3) b^3. 26. (x - x^2)^3 + (x^2 - 1)^3 + (1 - x)^3. (_M. I. T._) 27. (27y^3)^2 - 2(27y^3)(8b^3) + (8b^3)^2. (_Princeton._) 28. (a^3 + 8b^3)(a + b) - 6ab(a^2 - 2ab + 4b^2). (_M. I. T._) Solve by factoring: 29. x^3 = x. 30. z^2 - 4z - 45 = 0. 31. x^3 - x^2 = 4x - 4. ~Reference:~ The chapter on Factoring in any algebra. HIGHEST COMMON FACTOR AND LOWEST COMMON MULTIPLE Define H. C. F. and L. C. M. Find by factoring the H. C. F. and L. C. M.: 1. 3x^2 - 3x, 12x^2 (x^2 - 1), 18x^3 (x^3 - 1). 2. (x^2 - 1)(x^2 + 5x + 6), (x^2 + 3x)(x^2 - x - 6). (_Harvard._) 3. x^2 - y^2, x^2 + y^2, x^3 + y^3, x^6 + y^6, x^6 - y^6. (_College Entrance Board._) 4. x^3 + x^2 - 2, x^3 + 2x^2 - 3. (_Cornell._) 5. x^5 - 2x^4 + x^2, 2x^4 - 4x^3 - 4x + 6. (_Yale._) 6. x^2 + a^2 - b^2 + 2ax, x^2 - a^2 + b^2 + 2bx, x^2 - a^2 - b^2 - 2ab. (_Harvard._) 7. 2x^2 - x - 15, 3x^2 - 11x + 6, 2x^3 - x^2 - 13x - 6. (_College Entrance Board._) 8. (tv - v^2)^3, v^3 - t^2v, t^3 - v^3, v^2 - 2vt + t^2. Pick out the H. C. F. and the L. C. M. of the following: 9. 8(x^2 + y)^(32) (t^2 + z)^(19) (m - n^3)^(14), 12(x^2 + y)^(23) (t^2 + z)^(41) (m - n^3)^(17), 18(m - n^3)^(11) (x^2 + y)^(39) (t^2 + z)^(37). 10. 17ax^3 (y + z)^(10) (y - x)^(19) (x + z)^(27), 34a^2 x^4 (y + z)^(11) (y - x)^(21) (x + z)^(13), 51a^3 x^5 (y + z)^4 (x + z)^(32) (y - x)^(29).