9. 3 ax2 - 15 ax + 18 a, 6 a2x2 + 24 a2x — 126 a2.

11. a(a1)2, a2(a2 — 1), 2 a(a2 + 2 a − 3).

12. 5(x2 — y2), 10(x + y)2, 15(x − y)2(x+y).

13. h2-5h 14, h2 - 10 h + 21, h2 — 49.

14. 16(ab+b), 8 a(a2 — b2), 24 ab(a2 + 2 ab + b2).

15. m2(m − 1)2, m(m2 — 1), m3(m1n — n).

16. (y+4) (y2 — 16), y2 — y — 20, xy + 4 x − ay — 4 a.

ADDITION AND SUBTRACTION OF FRACTIONS

68. The process is the same as in arithmetic. If the fractions do not have the same denominator, they must be reduced to fractions which do have the same denominator. The lowest common denominator (1. c. d.) is obtained by finding the lowest common multiple of the given denominators.

1. Perform the indicated addition and subtraction:

The 1. c. d. is 6 a2x2. The reduction of the fractions to the 1. c. d. is effected as follows:

In practice, much of the work can be done mentally and need not be written down.

The 1. c. d. is 4 ab(a - b). Reduce the fractions to the l. c. d. :

The lowest common denominator is x(x2 - 1). Before reducing the last fraction to one with the denominator x(x2 – 1), it is a convenience to be able to write x - 1 in place of 1 - x. As 1 x differs from -1 only

in algebraic sign, we can do this, provided we change also the sign of the fraction or else change the sign of the numerator.

The former change is

The reduction to the lowest common denominator, x(x2 - 1), is effected

x (x2 − 1) + (x − 1) = x(x + 1), 1x(x+1)=

х

2x-2

x(x2 - 1)

x(x + 1) x(x2 – 1)

3x+2x 2- x2

4 x

x(x2-1)

x

(c — b) (a — c)

Here the factor (a - b) occurs twice, and both times with the same order of the letters. The same is true of (c—b). But in the first denominator occurs the factor (c — a), in the last denominator occurs (a—c). It is a convenience to have (c — a) in both denominators. Since (a — c) differs from (c — a) only in sign, we write in the last denominator (c — a) and at the same time change the sign of the fraction. We obtain

(a - b) (c − a)` (a - b) (c —b) (c—b) (ca)

The 1. c. d. = (a — b) (c — a)(c —b). Reducing to a common denominator and adding,

с

b

c-a

a-b

(a — b) (c — a) (c—b) (a - b)(c − a) (c − b) (a - b) (c — a) (c — b)

8.

4.

x 1 x+1

m

5.

y

6.

α b
+

a + b

b

x + y

bc

+

y + z

+

2+x

(y − z) (z — x)' (z − x) (x − y) ' (x − y) (y — z)

24.

1

1

2 х x2

2

2 x

+

x

1

3x

- 1

x2+3x+2

3

1

(a− 2)(a — 3) (3 − a)(a− 1) (1 − a) (2 − a)

70. The equation ax2 + bx + c = 0 is a complete quadratic equation. It is a quadratic equation because the highest power of the unknown x is the second; it is complete because it contains a term involving the unknown x to the first power and a term c (called the absolute term) which is free from x.