Example 4. Prove the identity

17(5x+3α)2-2(40x+27a)(5x+3a) = 25x2 - 9a2.

Since each term of the first expression contains the factor 5x+3a, the first side = (5x+3a){17 (5x+3α) − 2(40x+27a)}

7. a3-4a2b+Sab2 - 8b3, a3+4a2b+Sab2 + 8b3.

Find the continued product of

8. (a - b)2, (a+b)2, (a2+b2)2. 9. (1-x)3, (1+x)3, (1+x2)3.

10. a2 - 4a+3, a2 - a -2, a2+5a+6.

11. 3-y, 3+y, 9 −3y+ y2, 9+3y+y2.

12. 1+c+c2, 1-c+c2, 1-c2 + c1.

13. Divide a3(a + 2)(a2 - a − 56) by a2+7a.

14. Divide the product of x2+x−2 and x2+4x+3 by x2+5x+6. 15. Divide 3x2(x+4)(x2 −9) by x2+x-12.

16. Divide the product of 2x2+11a-21 and 3a2 - 20a -7 by

a2-49.

17. Divide (2a2 – a − 3)(3a2 –a – -2) by 6a2-5a -6. 18. Divide 26-7x3-8 by (x+1)(x2+2x+4).

Prove the following identities:

19. (a+b)3- (a - b)2(a+b) = 4ab(a+b).
20. c-d1- (c − d)3(c + d) = 2cd(c2 – d2).
21. (m-n)(m+n)3 − m1 + n1 = 2mn(m2 — n2).
22. (x+y)1 - 3xy(x + y)2 = (x+y)(x2+y3).
23. 3ab(a - b)2 + (a − b)1 = (a − b)(a3 -- b3).

MISCELLANEOUS EXAMPLES III.

1. Find the product of 1022-12-3x and 2x − 4+3x2.
2. If a = 1, b = − 1, c = 2, d = 0, find the value of
a2 - b2 b2 - cd c3-b3

+

+

a2+b2 2b2+ cd 3abc

3. Simplify 2[4x − {2y + (2x − y) − (x+y)}].

4. Solve the equations :

x-3 2-x 1-2x

(1)

5. Write down the square of 2x3 − x+5.

(2)

3x-4y = 25,
5x + 2y = 7.

6. Find the H.C.F. and L. C. M. of 3a2b3c, 12a1b2c3, 15a3b3c.

7. Divide a1+4b4 by a2-2ab+2b2.

8. Find in dollars the price of 5k articles at 8a cents each.

9.

Find the square root of 2a – 8x3 +24x2 - 32x+16.

12. A is twice as old as B; twenty years ago he was three times as old. Find their ages.

13. Simplify (1-2x) - {3-(4-5x)}+{6-(7-8x)}.

14. The product of two expressions is

6x1+5x3y+6x2y2+5xy3 +6y1,

and one of them is 2x2+3xy+2y2; find the other.

15. How old is a boy who 2x years ago was half as old as his father now aged 40?

16. Find the lowest common multiple of 2a2, 3ab, 5a3bc, 6ab2c, 7a2b. 17. Find the factors of

(1) x2-xy-72y2.

(2)

6x213x+6.

18. Find two numbers which differ by 11, and such that onethird of the greater exceeds one-fourth of the less by 7.

19. If a = 1, b = −1, c = 2, d = 0, find the value of

3

2x

1

х

20. Simplify x-y- {22-'y − 7 - (~ − 4 ) + (2 – 1x))

21. Solve the equations :

(1) (3x-8)(3x + 2) − (4x − 11)(2x + 1) = (x − 3)(x+7) ;

22. A train which travels at the rate of p miles an hour takes q hours between two stations; what will be the rate of a train which takes r hours?

27. Find the quotient when the product of b3+c3 and b3 – c3 · is divided by b3 – 2b2c +2bc2 – c3.

28. A, B, and C have $168 between them; A's share is greater than B's by $8, and C's share is three-fourths of A's. Find the share of each.

29. Find the square root of 9xo − 12x3 +22x2 + x2 + 12x + 4.

30. Simplify by removing brackets a2- [(b −c)2 - {c2 − (a − b)2}].

CHAPTER XVIII.

HIGHEST COMMON FACTOR.

144. DEFINITION. The highest common factor of two or more algebraical expressions is the expression of highest dimensions which divides each of them without remainder.

Note. The term greatest common measure is sometimes used instead of highest common factor; but this usage is incorrect, for in Algebra our object is to find the factor of highest dimensions which is common to two or more expressions, and we are not concerned with the numerical values of the expressions or their divisors. term greatest common measure ought to be confined solely to arithmetical quantities, for it can easily be shown by trial that the algebraical highest common factor is not always the greatest

common measure.

The

145. We have already explained how to write down by inspection the highest common factor of two or more simple expressions. [See Chap. XII.] An analogous method will enable us readily to find the highest common factor of compound expressions which are given as the product of factors, or which can be easily resolved into factors.

Example 1. Find the highest common factor of

4ca and 2cx3 +4c2x2.

It will be easy to pick out the common factors if the expressions are arranged as follows:

2cx3+4c2x2 = 2cx2(x+2c);

therefore the H.C. F. is 2cx2.

Example 2. Find the highest common factor of 3a2+9ab, a3 - 9ab2, a3+6a2b+9ab2.

Resolving each expression into its factors, we have

3a2+9ab=3a(a+3b),

a3 - 9ab2 = a(a + 3b)(a− 3b), a3+6a2b+9ab2 = a(a+3b)(a+35);

therefore the H. C.F. is a(a+3b).

146. When there are two or more expressions containing different powers of the same compound factor, the student should be careful to notice that the highest common factor must contain the highest power of the compound factor which is common to all the given expressions.

Example 1. The highest common factor of

x(α - x)2, a(α- x)3, and 2ax(a – x)5 is (a−x)2.

Example 2. Find the highest common factor of
ax2+2a2x+a3, 2ax2 - 4a2x - 6a3, 3(ax+a2)2.

Resolving the expressions into factors, we have
ax2+2a2x+a3 = a(x2+2ax+a2)

Therefore from (1), (2), (3), by inspection, the highest common

factor is a(x+a).

17. a3-36a, a3 +2a2-48a.

19. 2x2 - 9x+4, 3x2 – 7x – 20. ̄ 20.

21. 4m2 - 9m2, 6m3 – 5m2 – 6m, 6m4+5m3 - 6m2. 22. 3a4x3-8a3x3+4a2x3, 3a5x2 - 11a1x2 +6α3x2, 3a4x3+16a3x3 – 12α2x3.

18.

3a2+7a-6, 2a2+7a+3.

3c1+5c3 - 12c2, 6c5+7ca – 20c3.