2. Find the square of 4 a x y +7 abx. Ans. 16 a2 x2 y2 + 56 a2 b x2 y + 49 a2 b2 x2, 3. Multiply 7x+1 by 7x-1. Ans. 49 x2 1. 4. Required those two quantities whose sum is 3x+2a Ans. 2x and x + 2 a. and difference x 2 a. 5. Expand (x — 4)2. 6. Multiply 4 ab+ 3 by 4 ab- 3. 7. Find the square of 14 a2 b2 + 10 x2 y. 8. Find the square of 4 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 yo. 15. Find the product of a16 + 1, a3 + 1, aa + 1, a2 + 1, a+1, and a 1. Ans. a32 16. Find the product of a + b, a — b, and a2 b2. 1. ने 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 abxyz are a xa xbxxx yxz. 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. OPERATION. (axa bac) = a (x − b + c) nomial; and if we divide the polynomial by a, we obtain the other factor. Hence, As a is a factor of each term it must be a factor of the poly 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 a x2 y3. Ans. 6xy (112y + 3 ax y3). 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 (1+x). 4. Find the factors of 8 a2x2 + 12 a3 x1 — 4axy. Ans. 4 ax (2 a x + 3 a2 x3-y). 5. Find the factors of 5 x y2+ 25 a x5 — 15 x3 y3. Ans. 5 (y+ 5 a x2 - 3 y3). 6. Find the factors of 7 ax-8 by 14x2. 7. Find the factors of 4 x2 y2-28 x3 y 44 x y2. 8. Find the factors of 55 a2c 11 ac + 33 a2 cx. ac+33 9. Find the factors of 98 a2x2 - 294 a3 x2 y2. 10. Find the factors of 15 ab2 c d 9 a b2 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+2ab+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 Omitting the term that is equal to twice the product of the square roots of the other two, take for each factor the square root of each of the other two connected by the sign of the term omitted. 3. Find the factors of x2-2xy + y2. Ans. (xy) (x − y). 4. Find the factors of 4 ac2+12 a cd + 9 d2. . Ans. (2ac3 d) (2 ac + 3 d). 5. Find the factors of 14 x z + 4x2x2. Ans. (12 x 2) (1 − 2 x z). Ans. (3x-1) (3x-1). 7. Find the factors of 25 x2 + 60 x + 36. Ans. (7 ax) (7 a—x). 9. Find the factors of 16 y2 - 16 a2 y + 4 a*. 10. Find the factors of 12 a x + 4 x2 + 9 a2. 11. Find the factors of 6x+1+9x2. CASE III. 70. When a binomial is the difference between two squares. RULE. Take for one of the factors the sum, and for the other the difference, of the square roots of the terms of the bi - y = NOTE. When the exponents of each term of the residual factor obtained by this rule are even, this factor can be resolved again by the same rule. Thus, x - y = (x2 + y2) (x2 — y2); but x2 (x + y) (x − y); and therefore the factors of — y1 are x2 + y2, x+y, and x y. 9. Find the factors of a1 b. Ans. (a2 + b2) (a + b) (a — b). 10. Find the factors of x - y3. Ans. (x + y1) (x2 + y2) (x + y) (x − y). Ans. (1x) (1 + x2) (1 + x) (1 − x). 11. Find the factors of a1 12. Find the factors of 1 71. Any binomial consisting of the difference of the same powers of two quantities, or the sum of the same odd powers, can be factored. For |