THEOREM II. 58. The square of the sum of two quantities is equal to the square of the first, plus twice the product of the two, plus the square of the second. Let a and b represent the two quantities; their sum will be a+b. 59. The square of the difference of two quantities is equal to the square of the first, minus twice the product of the two, plus the square of the second. Let a and b represent the two quantities, and a > b; their dif ference will be a - b. 60. The product of the sum and difference of two quantities is equal to the difference of their squares. Let ab be the sum, and a b the difference of the two quantities a and b. 5. 3xy4ab by 3xy-4 ab. 61. This theorem suggests an easy method of squaring numbers. For, since a2 (a - b) (a + b) + b2, = (99 — 1) (99 + 1) + 12 = 98 × 100 + 1 = 9801. In like manner, 962: = 92 X 100+ 16 = 9216. 9982996 X 1000+ 4 = 996004. 4972 = 494 × 500 + 9 = 247 × 1000 + 9 = 247009. 2. Find the square of 4 a x y +7 abx. Ans. 16 a2x2 y2 + 56 a2 b x2 y + 49 a2 b2 x2. 3. Multiply 7x+1 by 7 x 1. Ans. 49 x2 - 1. 4. Required those two quantities whose sum is 3 x + 2 a and difference x Ans. 2x and x + 2 a. 2 a. 5. Expand (x2 — 4)2. 6. Multiply 4 ab+ 3 by 4 ab- 3. 7. Find the square of 14 a2 b2 + 10 x2 y. 10. Find the square of 3 ax-8axy. Aus. 9 a2x2-48 a2x2y +64 a2x2 y3. 11. Find the square of 2 a + b. 12. Find the value of (6 a + 4) (6 a 4) (36 a2 + 16). Ans. 9 at x2+ 24 a2 b x y3 + 16 b2 y®. 15. Find the product of a16 + 1, ao + 1, aa + 1, a2 + 1, a+1, and a Ans. a82 1. 1. 16. Find the product of a + b, ab, and a2 — b2, SECTION IX. FACTORING. 62. FACTORING is the resolving a quantity into its fac tors. 63. The factors of a quantity are those integral quantities whose continued product is the quantity. NOTE. In using the word factor we shall exclude unity. 64. A PRIME QUANTITY is one that is divisible without remainder by no integral quantity except itself and unity. Two quantities are mutually prime when they have no common factor. 65. The PRIME FACTORS of a quantity are those prime quantities whose continued product is the quantity. 66. The factors of a purely algebraic monomial quantity are apparent. Thus, the factors of a2bxyz are ахахъх х Ху Х z. 67. Polynomials are factored by inspection, in accordance with the principles of division and the theorems of the preceding section. CASE I. 68. When all the terms have a common factor. 1. Find the factors of a x ab + ac. nomial; and if we divide the polynomial by a, we obtain the other factor. Hence, RULE. Write the quotient of the polynomial divided by the common factor in a parenthesis, with the common factor prefixed as a coefficient. 2. Find the factors of 6 xy-72 x y + 18 ax2 y3. y2+18 Ans. 6xy (112y + 3 ax y2). NOTE. Any factor common to all the terms can be taken as well as 6 x y; 2, 3, x, y, or the product of any two or more of these quantities, according to the result which is desired. In the examples given, let the greatest monomial factor be taken. 3. Find the factors of x + x2. Ans. x (1x). 4axy. 4. Find the factors of 8 a2x2 + 12 a3 x1 Ans. 4 ax (2 a x + 3 a2 x3 — y). 5. Find the factors of 5 x y +25 a x5 - 15 x3 y3. Ans. 5 (y2+ 5 a x2 — 3 y3). 6. Find the factors of 7 a x-8 by + 14 x2. 7. Find the factors of 4 x2 y2-28 x3 y1 — 44 x1 y3. 8. Find the factors of 55 a2 c 11ac +33 a2 c x. 9. Find the factors of 98 a2x2 - 294 a3 x2 y2. 10. Find the factors of 15 abcd-9ab2 d2 + 18 a3 c2 d1. CASE II. 69. When two terms of a trinomial are perfect squares and positive, and the third term is equal to twice the product of their square roots. 1. Find the factors of a2 + 2 a b + b2. OPERATION. a2+2ab+b2 = (a + b) (a + b) principle in Theorem II. Art. 58. We resolve this into its factors at once by the converse of the |