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Example. Resolve into factors 7x2 – 19x – 6.

Write down (7x 3) (x 2) for a first trial, noticing that 3 and 2 must have opposite signs. These factors give 7x2 and – 6 for the first and third terms. But since 7x2 – 3x1=11, the combination fails to give the correct coefficient of the middle term.

Next try (736 2) (2 3).

Since 7x3 - 2x1=19, these factors will be correct if we insert the signs so that the negative shall predominate. Thus

7x2 – 19x – 6=17x + 2) (2 – 3). [Verify by mental multiplication.]

85. In actual work it will not be necessary to put down all these steps at length. The student will soon find that the different cases may be rapidly reviewed, and the unsuitable combinations rejected at once.

It is especially important to pay attention to the two following hints :

l. If the third term of the trinomial is positive, then the second terms of its factors have both the same sign, and this sign is the same as that of the middle term of the trinomial.

2. If the third term of the trinomial is negative, then the second terms of its factors have opposite signs. Example 1. Resolve into factors 14x2 + 29.x – 15 ......

..(1), 14x2 - 29x – 15

..(2). In each case we may write down (

73) (2x 5) as a first trial, noticing that 3 and 5 must have opposite signs.

And since 7 x 5 – 3 x 2 = 29, we have only now to insert the proper signs in each factor.

In (1) the positive sign must predominate,
in (2) the negative .
Therefore 14x2 +290 – 15=(7x – 3) (2x+5).

14.x2 – 292 – 15=(7x+3) (2x – 5). Example 2. Resolve into factors 5x2 +17x+6 ........ (1),

5x2 – 17x + 6

.(2). In (1) we notice that the factors which give 6 are both positive. In (2)

negative, And therefore for (1) we may write (5x + ) (x+ ). (2)

(5x – )(x-). And, since 5x3+1x2=17, we see that

5x2+17x+6=(5x + 2) (x+3).
5x - 17x+6=15x – 2) (x – 3).

Note. In each expression the third term 6 also admits of factors 6 and 1; but this is one of the cases referred to above which the student would reject at once as unsuitable. Example 3. 9x2 – 48xy +64y2 = (3x 8y) (3x – 8y)

= (3.c-8y)? Example 4. 6+70-5x2=(3+52) (2-2).

EXAMPLES XI. e.

Resolve into factors : 1. 2.02 + 3x +1. 2. 3x2 + 5x +2. 3. 2.02 +5.0 +2. 4. 3x2 + 10x +3. 5. 2.x2 +9x +4. 6. 3.x2 + 8x +4. 7. 2x2 +7x+6. 8. 2x2 + 11x +5. 9. 322 +11% +6. 10. 5x2 +11x+2. 11. 2.x2 + 3x – 2. 12. 3.02 +3 – 2. 13. 4.x2 +11x – 3. 14. 3.02 + 14.x – 5. 15. 2.x2 + 15x – 8. 16. 222 – X – 1. 17. 3.02 +7x – 6. 18. 2x2 + x - 28. 19. 3x2 + 13x – 30. 20. 6x2 + 7x – 3. 21. 6x2 – 7x - 3. 22. 3x2 + 7x +4. 23. 3x2 +23x + 14. 24. 2.x2 – X – 15. 25. 3.x2 + 19.x – 14. 26. 3.x2 – 19x – 14. 27. 6x2 – 31x +35. 28. 4x2 + x – 14. 29. 3x2 – 13x + 14. 30. 3.x2 +41x + 26. 31. 4x2 +23x+15. 32. 2.x2 5xy 3y2. 33. 8.x2 – 38x +35. 34. 12.x2 – 23xy +10y.

35. 15x2 +224x – 15. 36. 15x2 77x+10.

37. 12x2 – 31x -- 15. 38. 24.x2 + 22x – 21.

39. 72x2 – 145x+72. 40. 24x2 — 29xy 4y2.

41. 2 – 3x – 2.xo. 42. 3+11x - 4x4. 43. 6+ 5.x – 6x2. 44. 4-5x – 6x2. 45. 5+32.0 — 21x4. 46. 7+10x+3c”. 47. 18 - 33x +5.x4. 48. 8+6x – 5x2. 49. 20-9c - 2Ooo. 50. 24+37x — 72x2.

WHEN AN EXPRESSION IS THE DIFFERENCE OF Two

SQUARES. 86. By multiplying a+b by a b we obtain the identity

(a+b)(a-6)=a?-12, 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.

Example. Resolve into factors 25x2 - 16y?.

25x2 – 16y2=(5x)2 – (4y). Therefore the first factor is the sum of 5x and 4y, and the second factor is the difference of 5x and 4y.

.. 25x2 – 16y2=(5x+4y) (5x 4y). The intermediate steps may usually be omitted. Example. 1 – 49cR = (1+7c3) (1 – 7c»).

The difference of the squares of two numerical quantities is sometimes conveniently found by the aid of the formula

a? b2=(a+b) (a - b). Example. (329)2 – (171)2=(329+171) (329 – 171)

= 500 x 158
=79000.

EXAMPLES XI. f. Resolve into factors : 1. 2° –4. 2. a? – 81. 3. y2 – 100. 4. c2 – 144. 5. 9-a?. 6. 49-c. 7. 121 – x4 8. 400 - a? 9. 22 – 9a2. 10. 72 – 2522. 11. 36x2 – 256%, 12. 9.22 – 1. 13. 36p2 – 4992 14. 4k2 – 1.

15. 49 – 100K2. 16. 1-25x2

17. 42–462. 18. 9.x2 - y2. 19. poq- 36. 20. a-b2 – 4c-d2. 21. 24–9. 22. 9a4 - 121. 23. 25x2 – 64. 24. 8lat - 49.24. 25, 20 – 25.

26. 1-36a6. 27. 9.x4 – a?. 28. 81.26 — 25a. 23. x+a— 49. 30. a? – 64x6. 31. a2b2 – 9.x6. 32. My - 4 33. 1-a262, 34. 4- x4 35. 9 -4a2.

36. 9a4-2564. 37. 24–1662. 38. 22 – 25y

39. 1-10062. 40. 25 – 64x2. 41. 121–2 – 81x”. 42. paq-64a4. 43. 64.x2 — 2526. 44. 49c+ - 1674 45. 81p426 - 2562. 46. 16x16 – 9yo. 47. 36x36 – 49a14. 48. 1-100a6b4c2. 49. 25x10 – 16a8. 50. apb+c6 – x16.

Find by resolving into factors the value of 51. (575)2 – (425)2. 52. (121)2 – (120)2. 53. (750)2 – (250) 54. (339)2 – (319) 55. (753) - (253) 56. (101)-(99) 57. (1723)2-(277)

58. (1639)2 – (739)2. 59. (1811)2-(689).

60. (2731)2 – (269)2 61. (8133)2-(8131)

62. (10001)2 – 1.

87. When one or both of the squares is a compound quantity the same method is employed.

Example 1. Resolve into factors (a +20)2 – 16x2.
The sum of a +26 and 4x is a +26+4.x,

and their difference is a + 2b - 4.x.

.. (a +20)2 - 16x2 = (a +26+4x) (a +26 – 4x). Example 2. Resolve into factors x2 – (20 – 3c)2. The sum of x and 26 – 3c is x+ 25 – 3c, and their difference is x – (20 – 3c)=x – 20+ 3c.

.: x2 – (26 – 3c)2=(x+20 – 3c) (x – 20+3c). If the factors contain like terms they should be collected so as to give the result in its simplest form. Example 3. (3x + 7y) - (2x - 3y)

= {(3x + 7y) + (2x – 3y)} {(3x + 7y) - (2x – 3y)}
= (3x + 7y + 2x – 3y) (3x + 7y - 2x +3y)
=(5x+4y) (x2 +10y).

EXAMPLES XI. g. Resolve into factors: 1. (a+b)-- 2. (a - b)2–0%. 3. (1+y) - 4:4 4. (x+2y)2 - a?. 5. (a +36)2 - 16.x? 6. (x+5a) - 9ya. 7. (+ +5c)2-1. 8. (a -- 2.23)2 - 12 9. (2x - 3a)2 – 9c. 10. a-(6-0) 11. 22-(4+2). 12. 4a? (y-2). 13.9.2 - (2a-36)

14. 1-(a-6) 15. 02 (5a -36)

16. (a+b)2-(c+d)? 17. (a-b)-(x+y)

18. (7x+y)-1. 19. (a+)-(m-na.

20. (a-n)-(6+m) 21. (b-c)2-(a-x)

22. (4a+2)2-(6+y). 23. (a+26)2 – (3x+4y)2. 24. 1-(7a-30). 25. (a - b)2-(23-y).

26. (a-3.c)2 - 16y. 27. (2a-5.c)2 - 1.

28. (a+b-c)-(x-Y+2). 29. (3a+26)2-(c+x-2y).

Resolve into factors and simplify: 30. (x+y)2 – 24. 31. 22--(y x)? 32. (x+3y)2 4ya. 33. (24x+y)2 – (23.x - y)2. 34. (5x+2y)2 – (3x - y)2. 35. 9.22-(3.6 - 5y).

36. (7.2+3)2-(5.2-4)

37. (3a + 1)2 – (20 – 1)?. 38. 16ao – (3a + 1)”.
39. (2a+b-c)2-(a-b+c). 40. (x-7y+z)-(Ty-2)
41. (x+y-8)2 – (x – 8)2. 42. (2x+a-3-(3- 2.x)

88. By suitably grouping together the terms, compound expressions can often be expressed as the difference of two squares, and so be resolved into factors. Example 1. Resolve into factors a? – 2ax + x2 -- 462. a’ – 2ax + x2 – 462 = (a– 2ax + x2) – 462

=(a – 2)2 – (20)

=(a – x + 2b) (a - x - 20). Example 2. Resolve into factors 9a2 - c2 + 4cx – 4.x2. 9a? – c2 + 4cx – 4.x2=9a? – (c2 – 4cx + 4x2)

= (3a)2 – (0 – 2x)

=(3a +C – 2x) (3a - c+2x). Example 3. Resolve into factors 12xy + 25 – 4x2 – 9y2. 12xy + 25 – 4x2 – 9ya=25 – (4x2 – 12xy +9ya)

=(5)2 – (2x – 3y)?

=(5+ 2x – 3y) (5 - 2x + 3y). Example 4. Resolve into factors 26d - ao – c2 +12 + d2 + 2ac.

Here the terms 2bd and 2ac suggest the proper preliminary arrangement of the expression. Thus 2bd - ao – c2 +62 +d2 + 2ac=82 +2bd + d2 – a2 + 2ac – co

= 12 +26d+d2 – (a2 – 2ac +ca)
=(b+d)-(a - c)?
=(b+d+a-c) (6+d-a+c).

EXAMPLES XI. h.

Resolve into factors : 1. 22+ 2xy +ya- a.

2. a? – 2ab +62 – x2. 3. x2 – 6ax +9a2 – 1664.

4. 4a2 +4ab+62-9c. 5. x2 + a2 + 2ax - y2.

6. 2ay+a2 + y2 – 22. 7. 22 – Q? - 2ab - 62.

8. 72 c2 +2cx – 22. 9. 1 – 22 – 2xy - y2

10.02- -yo+2.cy. 11. x2 +y + 2xy - 4.xya. 12. a? — 4ab+462 – 9ac-. 13.12 + 2xy +y-al-2ab-b2 14. a? - 2ab +82-2-2cd-da.

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