38. x(x2+2x+2)+2(x2+2x+2), x1+4. 39. (2 c−y)4 c2 — (2 c− y)4 cy + (2 c−y)y2, (2 c− y)3 40. 6 a2+a-2, 90 a3-25 a2-10 a, 4 a2+2 a−2. 41. x+2x2+2x+1, x3+1. 42. Solve, using factoring: A square, 441 feet on a side, has a grass plot within it, 432 feet on a side. The remaining part of the square is a concrete walk. Find the cost of the walk at 14e per square foot. Additional work in factoring will be found in §§ 236 and 237. SOLUTION OF EQUATIONS BY FACTORING 103. The solution of equations affords an important and interesting application of factoring. Let it be required to solve the equation (x-3)(2x+5)=0. It is evident that the equation will be satisfied when x has such a value that one of the factors of the first member is equal to zero; for if any factor of a product is equal to zero, the product is equal to zero. Hence, the equation will be satisfied when x has such a value that either x-3=0, (1) (2) It will be observed that the roots are obtained by placing the factors of the first member separately equal to zero, and solving the resulting equations. 104. Examples. 1. Solve the equation x2-5x-24=0. Factoring the first member, (x−8)(x+3)=0. Placing the factors separately equal to 0 (§ 103), we have and x-8=0, whence x=8; x+3=0, whence x = −3. (§ 92) Verify by substituting x=8, x=-3 successively in the given equation. 2. Solve the equation 4 x2-2 x=0. Factoring the first member, 2 x(2 x −1)=0. Placing the factors separately equal to 0, we have 3. Solve the equation 3+4x2-x-4=0. Then, and Verify these results. x+4=0, whence x=- -4; x+1=0, whence x = -1; x-1=0, whence x=1. 4. Solve the equation x3-27— (x2+9 x−36)=0. Factoring the first member, we have by §§ 92 and 97, Placing the factors separately equal to 0, x=3, -3, or 1. Verify. The pupil should endeavor to put down the values of x without actually placing the factors equal to 0, as shown in Ex. 4. 9. 9 v2(2 v-3)-9 v(2 v-3)-4(2 v-3)=0. 10. 3x2-kx-4 k2=0. II. 10 u2-7u-12=0. 12. xz+2x-3z-6=0. 13. 15 v2+v-2=0 (Solve for x, then solve for k.) 14. 4 x3+20 x2-9x-45-0. 15. 28 t2-t-2=0. 16. 18x2-27 abx-35 a2b2=0. 17. n2+14 n-32=0. 18. x2+8x+16=0. 19. m3 +6 m2-9 m-54-0. 20. (x-3)-(3x+2)2=0. 21. 10 v2-39 v+14=0. 22. 15 x2+x-6=0. 23. (4 x2-49)(x2−3 x−10)(8 x2+14 x−15)=0. 24. (x-2)(5 x2+8x-4)-(x2-4)=0. 25. What number added to its square gives 30? 26. What number subtracted from 4 times its square gives? 27. If to 4 times the square of a certain number we add three times the number the result is 10. Find the number. 28. A rectangular room is 4 feet longer than it is wide, and its area is 96 square feet. What are its dimensions? Since we are finding dimensions of a room, these negative roots have no significance and can be rejected. There is, however, a very interesting geometrical interpretation which may be given. Consider 10 and Exercise 4. If measurement to the right is positive, then measurement to the left is negative. If distance upward is +, then distance downward is Now draw this rectangle : This gives two rectangles which fulfill the conditions of the problem, if one remembers that -12 is algebraically less than -8. +12 -12 +8 + -8 29. In a right triangle ABC, the base, AC, is 3 feet more than the altitude, BC, and the area is 14 square feet. Find AC and BC. Make a diagram with your results. 30. The perimeter of a rectangular field is 180 feet, and its area 1800 square feet. Find its dimensions. Make a diagram of your results. Find the equations whose roots (§ 73) are: 31. 2, -§. Subtracting each root from x, we have (x-2), (x-). By reversing § 103, the product of these expressions equated to zero gives the required equation. 41. The sides of a rectangle are 8 and 11. Form a problem similar to problem 28. State the equation. QUERIES 1. Is 2 a a number? Is it a sum? Is it a product? What are its factors? 2. Is a+ba number? Is it the sum of two numbers? Can you factor it? 3. Translate a2+b2 into English. Can you factor it? 4. Given two numbers F and S; if their sum is multiplied by their difference, what is the result? 5. Given two numbers F and S; if their sum be multiplied by itself, what is the result? Express in English. 6. Does the definition of division bear any relation to your idea of the process of factoring? 7. Is 4 a2+2a+1 a perfect square? Why? 8. The following are for mental drill: (304)2=? (20})2=? (294)2= ? 9. Is 3 a root of the equation 3 x2-4x+7=0? Why? Is x-3 a factor of the expression? 10. Is 2 a root of 2 m2-9m+10=0? Is m-2 a factor of the expression? 11. How do you form the equation whose roots are 3 and 7? 12. If one root, 5, of x2-8x+15=0 is given, can you find the other root without solving the equation? 13. Using your knowledge of § 91, can you make a general statement covering the results of examples 13 and 14, Exercise 12? VIII. HIGHEST COMMON FACTOR. LOWEST COMMON MULTIPLE (We consider in the present chapter the Highest Common Factor and Lowest Common Multiple of Monomials, or of Polynomials which can be readily factored by inspection. The Highest Common Factor and Lowest Common Multiple of polynomials which cannot be readily factored by inspection, will be considered in a more advanced course in algebra.) HIGHEST COMMON FACTOR 105. The Highest Common Factor (H. C. F.) of two or more expressions is their common factor of highest degree (§ 58). If several common factors are of equally high degree, it is understood that the highest common factor is the one having the numerical coefficient of greatest absolute value in its term of highest degree. For example, if the common factors were 6 x and 2 x, the former would be the H. C. F. 106. Two expressions are said to be prime to each other when unity is their highest common factor. 107. CASE I. Highest Common Factor of Monomials. Ex. Required the H. C. F. of 42 a3b2, 70 a2bc, and 98 a1b3d2. By the rule of Arithmetic, the H. C. F. of 42, 70, and 98 is 14. It is evident by inspection that the expression of highest degree which will exactly divide a3b2, a2bc, and a b3d2 is a2b. Then, the H. C. F. of the given expressions is 14 ab. It will be observed, in the above result, that the exponent of each letter is the lowest exponent with which it occurs in any of the given expressions. |