(3) x2+4x3y+6 x2y2 + 4 xy3 + ya.

x2 + 4x3y +6 x2y2 + 4 xy3 + y1

=

= (x4 + 4x3y + 3 x2y2) + (3x2y2 + 4 xy3 + y1)
x2 (x2+4 xy + 3 y2) + y2 (3 x2 + 4 xy + y2)
= x2(x+3y) (x + y) + y2 (3 x + y) (x + y)

(4) 23 +5 x1y + 10 x3y2 + 10 x2y3 + 5 xy1 + y3

=

=

(x + y3) + (5 x1y + 5 xy1) + (10 x3y2 + 10 x2y3)
(x + y) (x − x3y + x2y2 — xy3 + y1)

+5 xy (x + y) (x2 − xy + y2) + 10 x2y2 (x + y) = (x + y) (x* + 4 x3y +6 x2y2 + 4 xy3 + y1)

· (x + y) (x + y) (x + y) (x + y) (x + y).

(5) 8x-12x2+6x-1

=

=

=

(2x-1) (4x2-4x+1)=(2x-1) (2 x − 1) (2 x −1).

5. a-5 a 10 a3 - 10 a2 + 5 a −1.

6. 8 a3 +36 a2b + 54 ab2 + 27 b3.

7. x3а -3x2а + 3 xa − 1.

8. 2727 a +9 a2 + a3.

9. x+4x3y1 +6 x 2y2+ 4x ̄1y-3 + y^.

30. (a+b) (a — b)*.

31. (x+y) (x2 — y2) — (x − y) (x2 + y2).

32. 49x14x2+1.

34. x-4x+6x2-4x+1.

40. 2+y+5 (x1y + xy1) + 10 (x2y3 + x3y3).

43. (x+y)3 + (x − y)3 + (x2 + y2)3 — 3 (x1 — y1).

44. 22+ y2+z2 + 2xy + 2 xz + 2 yz.

45. 4a2+962 +16 c2 + 12 ab-16 ac- 24 bc.

46. 4x-4x+1-16 y2-8 yz - z2.

60. a1- 82 a2b2 + 81 b1.

61. 27 a3+64 b3+ 108 ab + 144 ab2.

62. (a2+b2 + c2)2 — 4 a2b2.

63. (3x2-2xy - 5 y2)+(6 x2-9 xy-15 y2).

64. 81 (a + b + c)2 − 36 (x + y)2.

76. (xy)-11(x + y) + 18.

77. (x + y)2-44(x + y) (a + b) + 43(a+b)2.

78 x2n - y2n

79. a

-2x-b-2x.

80. a2n+12

1.

81. 2a+12 y2a+12.

36

3c

82. Qua t3+23_3g.

83. 5(x+y)2+7(x2 — y2).

84. (a2-2a+3)-(a2-3a-4)2.

90. (a+b)1+(a+b)2 (a − b)2+(a — b)*.

91. (a2 + b2)2− 1 + (a2 + b2 + 1).
92. x2+ y2+22-2xy - 2xz + 2 yz.

93. (x4 +423 +3x2)+(3x2 + 4 x + 1).
94. 3(2+)+1+

95. a2(m+n) + 2 am+nb + b2.

96. a3 +27 b3 + 125 c3 — 45 abc.

CHAPTER VII.

HIGHEST COMMON FACTOR.

145. Common Factors. A common factor of two or more expressions is an expression which will exactly divide each of them.

Thus, 5 x is a common factor of 15a2 and 25 ax2.

146. Highest Common Factor. The highest common factor of two or more expressions is the common factor of highest degree which will exactly divide each of them.

Thus, 5 x2 is the highest common factor of 15x2 and 25 ax2.

147. When unity is the highest common factor of two expressions, they are said to be prime to each other.

Thus, a2, x, 5, having no factor common except unity, are prime to each other.

148. H. C. F. stands for "Highest Common Factor."

149. I. When the factors can be determined by inspection. Find the H. C. F. of the following:

(1) 25 ab2 and 75 ab2z.

25 ab2= 5 x 5 x axbxb;

75 ab2%= 5 x 5 x 3 xa xbx b xz;

.. H. C. F. is 5 × 5 × a × b × b = 25 ab2.

(2) 96 m2n, 144 mm2, 1728 m2n2.

96 m2n = 3 × 2 × 2 × 2 × 2 × 2 × m × m × n;
3 × 3 × 2 × 2 × 2 × 2 × m × n × n;

144 mn2 1728 m2n2

=

=

3 × 3 × 3 × 2 × 2 × 2 × 2 × 2 × 2 × m × m x n x n :

.. H. C. F. = 3 × 2 × 2 × 2 × 2 × m × n = 48 mn.