Let PX QX R=0 (1) be the given equation. Then we are to prove that the equation is equivalent to the set of equations For any solution of (1) must reduce Px Q × R to 0, and, therefore, by Ch. III., § 3, Art. 18, either P, or Q, or R to 0. That is, every solution of (1) is a solution of one of equations (2). Any solution of P = 0 must reduce P to 0, and, therefore, by Ch. III., § 3, Art. 16, P Q R to 0. That is, every solution of P = 0 is a solution of P × Q × R = 0. In like manner, it can be shown that the solutions of Q = 0 and R = 0 are solutions of (1). Ex. Solve the equation x(x-2)(x+5)= 0. The given equation is equivalent to the equations = 2. The expression (x − 1)(x − 2) reduces to 0 for x = 1 and x = 2, and the expression (x − 5) (x + 4) reduces to 0 for x = 5 and x 4. Observe that the two expressions do not have a common factor, and do not reduce to 0 for the same values of x. This example illustrates the following principle: If two expressions in one and the same unknown number do not have a common factor, they cannot reduce to 0 for the same value of the unknown number. Let E1 and E2 be two integral expressions in x which do not have a common factor. Then we are to prove that E1 and E2 cannot reduce to 0 for the same value of x. For if E1 and E2 do reduce to 0 for the same value of x, say a, they must both be divisible by x a without a remainder (Ch. VI., § 2, Art. 4). That is, x - a must be a factor of both expressions. But this contradicts the hypothesis that E1 and E2 do not have a common factor. CHAPTER IX. FRACTIONS. 1. The quotient of a division can be expressed as an integer or an integral expression only when the dividend is a multiple of the divisor; as a2b ÷ ab = a; (ax2 + 2 bx) ÷ x = ax + 2b. If the dividend be not a multiple of the divisor, the quotient is called a Fraction; as a÷b; (ax2 +2 bx) ÷ x3. 2. The notation for a fraction in Algebra is the same as in ordinary Arithmetic. ax2 + 2 bx Thus, (a+2 bx) is written. 2-3 The Solidus,/, is frequently used instead of the horizontal line to denote a fraction; as (ax2 + bx)/ for ax2 + bx. bx)/x3 3. As in Arithmetic, the dividend is called the Numerator of the fraction, the divisor the Denominator, and the two are called the Terms of the fraction. 4. An integer or an integral expression can be written in a fractional form with a denominator 1. It is important to notice that an algebraic fraction may be arithmetically integral for certain values of its terms. E.g., when a = 4 and b = 2, the fraction a/b becomes 4/2 = 2. 5. By the definition of a fraction, a/b is a number which, multiplied by b, becomes a; that is, xb = a (a/b) × b = a, or 2 × b = (1) 6. The Sign of a Fraction. The sign of a fraction is written before the line separating its numerator from its denominator; Since a fraction is a quotient, the sign of a fraction is determined by the rule of signs in division. (i.) If the signs of the numerator and the denominator of a fraction be reversed, the sign of the fraction is unchanged. This step is equivalent to multiplying or dividing both terms of the fraction by - 1. (ii.) If the sign of either the numerator or the denominator of a fraction be reversed, the sign of the fraction is reversed; and conversely. (iii.) If the signs of an even number of factors in the numerator and denominator, either or both, of a fraction be reversed, the sign of the fraction is unchanged; but, if the signs of an odd number of factors be reversed, the sign of the fraction is reversed. 8. Observe that the sign of a fraction affects each term of the numerator (or each term of the denominator); or, the dividing line between the numerator and the denominator has the same effect as parentheses. Change each of the following fractions into an equivalent fraction with sign reversed, leaving the denominator unchanged: Change each of the following pairs of fractions into two equivalent fractions whose denominators are equal: Change each of the following pairs of fractions into two equivalent fractions whose denominators have a common factor: 9. A Proper Fraction is one whose numerator is of lower degree than its denominator in a common letter of arrange ment. E.g., x-2 1 An Improper Fraction is one whose numerator is of the same or of a higher degree than its denominator in a common letter of arrangement. x2+3x2 + x 1 A Fractional Expression is an expression which has one or more fractional terms. If both integral and fractional terms occur in an expression, it is sometimes called a Mixed Expression. An improper fraction can be reduced to a mixed expression. Thus, if x2-4x+3 be divided by +2x-1, the quotient will be x-1, and the remainder +2. Therefore, by Ch. III., § 4, Art. 9, Reduce each of the following fractions to equivalent fractional expressions, containing only proper fractions: 10. The reduction of fractions is based upon the following principles: (i.) If both numerator and denominator of a fraction be multiplied by one and the same number or expression, not 0, the value of the fraction is not changed; or, stated symbolically, (ii.) If both numerator and denominator of a fraction be divided by one and the same number or expression, not 0, the value of the fraction is not changed; or, stated symbolically, |