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 7× 2-3x1=11, the combination fails to give the correct coefficient of the middle term.

Next try (7x 2) (x 3).

Since 7×3-2x1=19, these factors will be correct if we insert the signs so that the negative shall predominate.

Thus

7x2-19x-6=(7x+2) (x − 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:

1. 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+29x-15.........

.(1),

14x2-29x- 15. ...........(2).

In each case we may write down (7x 3) (2x 5) as a first trial, noticing that 3 and 5 must have opposite signs.

And since 7 × 5 −3×2=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+29x-15=(7x− 3) (2x+5).
14x2-29x-15=(7x+3) (2x − 5).

.(1),

Example 2. Resolve into factors 5x2+17x+6.

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

In (2)

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

(2)

(5 – ) (

And, since 5 x3+1x2=17, we see that

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

).

negative.

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)

= =(3x-8y)2.

Example 4. 6+7x-5x2=(3+5x) (2−x).

38. 24x2+22x – 21.

40. 24x2-29xy – 4y2.

42. 3+11x-4x2. 43. 45.

41. 2-3x-2x2.

6+5x-6x2. 44. 4-5x-6x2.

7+10x+3.x2. 47.

18-33x+5x2.

5+32.x-21x2. 46. 48. 8+6x-5x2. 49. 20-9x-20x2. 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−b)=a2 — b2,

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.

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.

1-49c6=(1+7c3) (1 − 7c3).

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

a2-b2=(a+b) (a - b).

Example. (329)2 - (171)2=(329+171) (329 – 171)

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

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

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

.. (a+2b)2 - 16x2 = (a+2b+4x) (a+2b −4x).

Example 2. Resolve into factors x2-(2b - 3c)2.

The sum of x and 2b - 3c is x+2b-3c,

and their difference is x − (2b-3c)=x-2b+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)2 - (2x-3y)2

= {(3x+7y)+(2x-3y)} {(3x+7y)-(2x-3y)}

=(3x+7y+2x-3y) (3x + 7y - 2x+3y) =(5x+4y) (x+10y).

EXAMPLES XI. g.

Resolve into factors:

1. (a+b)2 —c2.

2. (a - b)2 - c2.
5. (a+3b)2-16x2.
8. (a-2x)2-b2.
11. 2-(y+2)2.

4. (x+2y)3-a2. 7. (x+5c)2-1. 10. a2 (b-c)2. 13. 9x2-(2a-3b)2. 15. c2-(5a-36)2. 17. (a-b)-(x+y)2. 19. (a+b)2- (m-n)2. 21. (b-c)2 - (a — x)2. 23. (a+26)2 - (3x+4y)2.

25. (a-b)2-(x-y)2.

27. (2a-5x)2-1.

29. (3a+26)2- (c+x−2y)2.

3.

(x+y)2 - 4%2.

6. (x+5a)2-9y2. 9. (2x-3a)2-9c2. 12. 4a2 — (y — z)2.

14. 1-(a-b)2.

Resolve into factors and simplify:

30. (x+y)2x2. 31. x2-(y-x)?. 32. (x+3y)2 - 4y2.

33. (24x+y)-(23x-y)2.

35. 9.x2-(3x- 5y)2.

34. (5x+2y)2- (3x-y)2.

36. (7x+3)2- (5x-4)2.

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 a2 - 2ax + x2 − 4b2.

a2 - 2ax + x2 - 4b2 = (a2 — 2ax + x2) − 4b2

Example 2. Resolve into factors 9a2 - c2+4cx - 4x2.
9a2-c2+4cx-4x2=9a2 - (c2 - 4cx+4x2)
= (3a)2 - (c-2x)2

=(3a+c−2x) (3a − c+2x).

Example 3. Resolve into factors 12xy + 25 - 4x2 - 9y2. 12xy +25 - 4x2 - 9y2=25 - (4x2 - 12xy + 9y2)

Example 4. Resolve into factors 2bd-a2 - c2+b2+ d2 + 2ac.

Here the terms 2bd and 2ac suggest the proper preliminary arrangement of the expression.

Thus

2bd-a2-c2+b2+d2+2ac = b2+2bd+d2- a2+2ac - c2

= b2+2bd+d2 — (a2 — 2ac + c2)

=(b+d)2- (ac)2

=(b+d+a−c) (b+d−a+c).