Page:Elementary algebra (1896).djvu/96

78 Rh 6. x^+x-6. 20. x^ + 23rK+ 102. 6. 2 ax^ + 3 axy - 2 bxy - 3 by^. 21. amx^2 + bmxy - anxy - bny^2. 7. x^+7xy -60y^. 22. 6x^2 -7x -3. 8. a^2 - ay - 210 y^2. 23. 3 + 11 x - 4 x^. 9. x^2 - 21 x + 110. 24. 12x^2 - 23xy + 10y^2. 10. 24 + 37 x - 72 x^2. 25. 3 x^2 + 7 x + 4. 11. 98-7x-x^2. 26. a^2 - 32^ + 256. 12. 3 .x^2 + 23 X 4- 14. 27. 3 x^2 - 19 x - 14. 13. 2 x^2 -f 3 X - 2. 28. x^2 - 19x + 90. 14. x^2 -20xy - 96 y^2. 29. x^2 + 3 x - 40. 15. a^2 - 20 abx + 75 b^x^. 30. x^2y^2 - 16xy + 39. 16. a^2 - 24 a + 95. 31. 204 - 5 x - x^2. 17. 7 + 10 x + 3 x^2. 32. 15 x^2 + 224 x - 15. 18. a^2 - 4a - 21. 33. 3x^2 + 41 x + 26. 19. x^2 + 43 xy + 390 y^3 34. 65 + 8 xy - x^2y^2.

WHEN AN EXPRESSION IS THE DIFFERENCE OF TWO SQUARES.

98. By multiplying a +b by a - b we obtain the identity (a + b)(a - b) = a^2 - b^2

a result which may be verbally expressed as follows :

The product of the sum and the difference of any two quantities is equal to the difference of their squares.

Conversely, the difference of the squares of any two quantities is equal to the product of the sum and the difference of the two quantities.

Ex. 1. Resolve into factors 25x^2 - 16 y^2.

25 x^2 - 16 y^2 = (5 x)^2 - (4 y)^2.

Therefore the first factor is the sum of 5x and 4y, and the second factor is the difference of 5x and 4 y. .: 25x^2- 16 y^2 =(5x+ 4 y) (5 x- 4y). The intermediate steps may usually be omitted.

Ex.2. 1 -49^6=(1 + 7c^3)(1 - 7c^3).