Putting -b in place of b in (1), we obtain a" - (-b)" =a" 1+a" 2(-b) + ··· + (-b)"'. a-(-b) -1 ... = If n is even, (b)" = b", and (— b)" -1 — — b′′-1 (Art. 109) 1 If n is odd, (— b)" — — b", and (− b)” -1 = + b′′-1. Formulæ (2) and (3) are in accordance with the laws stated in the first part of the article. 117. Write by inspection the values of the following; m, n, p, and q being positive integers: 1. (10 ab+9ac)2. 2. (13x-5x2)2. 3. (-7+2x). 4. (6abc)2. VII. FACTORING. 118. To resolve an algebraic expression into factors (Art. 62), is to find two or more expressions which, when multiplied together, will produce the given expression. In the present chapter we shall consider only the resolution of rational and integral expressions (Art. 23) into rational and integral factors. It is not always possible to resolve a rational and integral polynomial into rational and integral factors; but there are certain forms which can always be resolved. Only the more elementary methods will be considered in succeeding articles. 119. CASE I. When the terms of the expression have a common monomial factor. Example. Factor 14 xy-35 x3y2. It is evident by inspection that each term contains the monomial factor 7 xy2. Dividing the expression by 7xy, we have 2y-5x2. Whence, 14 xy-35xy=7xy2 (2 y2 -5x2). 120. CASE II. When the expression is the sum of two or more binomials which have a common binomial factor. 1. Factor am - bm + an — bn. By Case I., (am-bm) + (an-bn) = m (a−b)+n(a−b). The two binomials have the common factor (a — b). Dividing by ab, the quotient is m+n. Whence, am - bm + an - bn = (a — b) (m +n). Note. If the third term of the given expression is negative, it is convenient to enclose the last two terms in a parenthesis preceded by a - sign. 2. Factor 3 - 2x2 - 3x + 6. x3-2x-3x+6= (x3-2x2)-(3x-6) = x2(x-2)-3(x-2) = (x-2) (x2-3). 121. If an expression when raised to the nth power (n being a positive integer), is equal to another expression, the first expression is said to be an nth Root of the second. Thus, if a" = b, then a is an nth root of b. 122. The Radical Sign, √, when written before an expression, indicates some root of the expression. Thus, Va indicates a second, or square root of a1; 3 Va indicates a third, or cube root of a3; 2n Va indicates an nth root of a2; etc. The index of a root is the number written over the radical sign to indicate what root of the expression is taken; thus in Va, the index is n. If no index is expressed, the index 2 is understood. Note. It will be shown hereafter (Art. 667) that an expression has two different square roots, three different cube roots, and, in general, n different nth roots. It will be understood throughout the remainder of the work, unless the contrary is specified, that when we speak of "the square root," "the cube root," etc., we simply mean "one of the square roots," "one of the cube roots," etc. 123. A rational and integral expression is said to be a perfect power of the nth degree when it has a rational and integral nth root. 124. We will now give a rule for finding any root of a rational, integral, and positive monomial, which is a perfect power of the same degree as the index of the required root. 1. Required the square root of 9 a2b1. By Art. 109, (3 ab2)2 = 9 a2b1. Whence by Art. 121, √9 a2b1=3ab2. Note. We also have (−3 ab2)2 = 9a2b4; whence, √9 a2b1 = −3 ab2. A negative value of a root will not be considered in the examples of the present chapter; and when we speak of extracting "the root" of a positive monomial, the positive root will be understood. 2. Required the cube root of 8xy12. By Art. 109, Whence, (2x)=8xy12. We then have the following rule: Extract the required root of the numerical coefficient, and divide the exponent of each letter by the index of the root. It is evident from the above illustrations that a rational, integral, and positive monomial is a perfect power of the nth degree (Art. 123) when its numerical coefficient is a perfect power of the nth degree, and the exponent of each letter a positive integer divisible by n. 125. It follows from Art. 108, (1) and (2), that a rational and integral trinomial is a perfect square when its first and last terms are perfect squares and positive, and the second term plus or minus twice the product of their square roots. Thus, 4x-12xy2+9y' is a perfect square. 126. To find the square root (Art. 122, Note) of a perfect trinomial square, we simply reverse the rules of Art 108: Extract the square roots (Art. 124, Note) of the first and last terms, and connect the results by the sign of the second term. Thus, let it be required to find the square root of 4x2-12xy2+9y'. The square root of 4a2 is 2x, and of 9y' is 3y2. |