Ex. 2. Resolve into factors x^ - (2b - 3c)^2.
The sum of x and 2b -3c is x + 2 b - 3 c, and their difference is
x-(2b-3c)=x-2b + 3c. .-. x^2 - (2 b - 3 c)^2 = (x + 2 b - 3 c) (x - 2 5 + 3 c).
If the factor3contain like terms, they should be collected so as to give the result in its simplest form.
Ex.3. (3x + 7y)^2-(2x-3y)^2 = {(3x + 7 y) + (2x - 3y)}{(3x + 7 y)-(2x - 3y)} = (3x + 7y + 2x-3y)(3x+7y-2x + 3y) = (5x + 4y)(x + 10y).
EXAMPLES X. h.
Resolve into factors :
1. (a + b)^2 - c^2. 4. (x + 2 yy - a 7. (x + 5 c)^2 - . 2. (a-b)^2-c^2. 5. (a + 3b)^2-16x^2. 8. (a-2x)^2-62. 3. (x + y)^2-4z^2. 6. (x+ 5a)^2-9y^2. 9. (2x-3a)^2- 9c^2. 10. 9x^2 -(2a- 3b)^2. 18. (b-c)^2-(a-x)^2. 11. 1 - (a - b)^2. 19. (4 a + x)^2 - (b+y)^2, 12. c^2 - (5 a - 3 b)^2. 20. (a + 2 b)^2 - (3 x + 4 y)^2. 13. (a + b)^2 -(c + d)^2. 21. 1-(7a-3b)^2. 14. (a - b)^2 - (x + y)^2. 22. (a - b)^2 - (x - y)^2. 15. (7x + 2y)^2-1. 23. (a-3x)^2-16y^2. 16. (a + b)^2 - (m - n)^2 24. (2 a - 5 x)^2 - 1. 17. (a - n)^2 - (b + m)^2. 25. (a + b- c)^2 - (x - y + z)^2.
Resolve into factors and simplify:
26. (x + y)^2 - x^2. 27. x^2 -(y - x)^2. 28. (x + 3 y)^2 - 4 y^2. 29. (24 x + y)^2- (23 x - y)^2. 34. 16 a^2 - (3 a + 1)^2. 30. (5x + 2y)^2-(3x-y)^2. 35. (2a + b - c)^2 -(a - b + c)^2. 31. 9x^2-(3x-5y)^2. 36. (x -7 y + z)^2 -(7 y - z)^2. 32. (7 X + 3)^2 - (5x - 4)^2. 37. (x 4- y - 8)^2 - (x - 8)^2. 33. (3a + 1)^2-(2a-1)^2. 38. (2x + a - 3)^2 -(3 - 2x)^2.
100. Compound Expressions Arranged as the Difference of two Squares. By suitably grouping the terms, compound expressions can often be expressed as the difference of two squares, and so be resolved into factors.
