133. When a binomial is the sum or difference of two cubes. Therefore, the sum of two perfect cubes is divisible by the sum of their cube roots, and the difference of two perfect cubes is divisible by the difference of their cube roots. 1. Resolve into factors 8 a3 + 27 66. The cube root of 8 a3 is 2a, and of 27 66 is 362. (2a)3 + (362)3 = (2a + 362) (4 α2 6 ab2 +964). 2. Resolve into factors 125 x3 1. The cube root of 125 x3 is 5x, and of 1 is 1. By putting 5x for a and 1 for b in (2), we have 125 x3 1 = (5 x - 1) (25 x2 + 5 x + 1). 3. Resolve into factors x + y2. The cube root of x is x2, and of yo is y3. By putting x2 for a and y3 for b in (1), we have x® + y = (x2 + y3) (x4 — x2y3 + y). 4. Resolve into factors (x - y)2 + 28. The cube root of (x - y) is x y, and of z3 is z. By putting x - y for a and z for bin (1), we have 2 (x - y)3 + z = [(x - y) + z] [(x - y)2 - (x - y) z + z2] = (x - y + z) (x2 2 xy + y2 xz + yz + z2). xn - yn = xn - xn-1 y + xn-1 y - yn. Taking out xn-1 from the first two terms of the right side, and y from the last two terms, we have Now x of xn-1 y is an exact divisor of the right side, if it is an exact divisor yn-1; and if x - y is an exact divisor of the right side, it is an exact divisor of the left side; that is, x - y is an exact divisor of xn yn if it is an exact divisor of xn - 1 - yn-1. But xy is an exact divisor of x3 y3 (§ 117), therefore it is an exact divisor of x2 - y; and since it is an exact divisor of x y, it is an exact divisor of x5 y5; and so on, indefinitely. The method employed in proving this Theorem is called Proof by Mathematical Induction. 135. The Factor Theorem. If a rational and integral expression in x vanishes, that is, becomes equal to 0, when r is put for x, then x r is an exact divisor of the expression. By subtracting (2) from (1), the given expression assumes the form a (xn - rn) + b (xn − 1 — rn-1) + ..... + h (x - r). But x r is an exact divisor of xn γη, χη-1 - γηn-1, and so on. Therefore, x r is an exact divisor of the given expression. NOTE. If x - ris an exact divisor of the given expression, r is an exact divisor ofk; fork, the last term of the dividend, is equal to r, the last term of the divisor, multiplied by the last term of the quotient. Therefore, in searching for numerical values of x that will make the given expression vanish, only exact divisors of the last term of the expression need be tried. 1. Resolve into factors x3 + 3x2 - 13 х 15. The exact divisors of 15 are 1, -1, 3, — 3, 5, — 5, 15, 15. 13 x If we put 1 for x in x3 + 3x2 15, the expression does not vanish. If we put -1 for x, the expression vanishes. Therefore, x - (-1), that is, x + 1, is a factor. x2 + 3 x2 13 x 15 = (x + 1) (x2 + 2x – 15) = (x + 1) (x - 3) (x + 5). NOTE. An expression can sometimes be resolved into three or more factors. 2. Resolve into factors x2 - 26 х 5. By trial we find that the only exact divisor of 5 that makes the expression vanish is -5. Therefore, divide by x + 5, and we have x3 26 x - 5 = (x + 5) (x2 — 5 х — 1). As neither + 1 nor x2 1, the exact divisors of - 1, will make 5 x 1 vanish, this expression cannot be resolved into factors. |