EXAMPLES. 1. The composite numbers under 100, that is, 4, 6, 8, &c., may be given as examples. Every pupil should learn to give the factors of these quantities readily. 2. What are the prime factors of 105?. Ans. 3, 5, 7. Ans. 2, 3, 5, 7. 3. What are the prime factors of 210?. 4. Resolve 4290 into its prime factors.. Ans. 2, 3, 5, 11, 13. ART. 89.—A prime quantity, in Algebra, is one which is exactly divisible only by itself and by unity. Thus, a, b, and b+c are prime quantities; while ab and ab+ac are not prime. ART. 90.-Two quantities, like two numbers, are said to be prime to each other, or relatively prime, when no quantity except unity will exactly divide them both. Thus, ab and cd are prime to each other. ART. 91.-A composite number, or a composite quantity, is one which is the product of two or more factors, neither of which is unity. Thus, ax is a composite quantity, of which the factors are a and x. REMARK.- A monomial may be a composite quantity, as ax; and a polynomial may not be a composite quantity, as a2+x2. ART. 92.-To separate a monomial into its prime factors. RULE. Resolve the coefficient into its prime factors; then these, with the literal factors of the monomials, will form the prime factors of the given quantity. The reason of this rule is self-evident. Find the prime factors of the following nominals: ART. 93. To separate a polynomial into its factors, when one of them is a monomial and the other a polynomial. REVIEW.-87. What is the divisor of a quantity? 88. What is a prime number? What is a composite number? Name several of the prime numbers, beginning with unity. Name several of the composite numbers, beginning with 4. What is the rule for resolving any composite number into its prime factors? 89. What is a prime quantity? Give an example. 90. When are two quantities prime to each other? Give an example. 91. What is a composite quantity? Give an example. 92. What is the rula for separating a monomial into its prime factors? RULE. Divide the given quantity by the greatest monomial that will exactly divide each of its terms. Then the monomial divisor will be one fac tor, and the quotient the other. The reason of this rule is selfevident. Separate the following expressions into factors: ART. 94. To separate a quantity which is the product of two or more polynomials, into its prime factors. No general rule can be given, for this case. When the given quantity does not consist of more than three terms, the pupil will generally be able to accomplish it, if he is familiar with the theorems in the preceding section 1st. Any trinomial can be separated into two binomial factors, when the extremes are squares and positive, and the middle term is twice the product of the square roots of the extremes. See Articles 79 and 80. Thus: a2+2ab+b2=(a+b)(a+b). a2-2ab+b2 (a—b)(a—b). 2d. Any binomial, which is the difference of two squares, can be separated into two factors, one of which is the sum, and the other the difference of the roots. See Art 81. Thus: a-b(a+b)(ab). 3d. When any expression consists of the difference of the same powers of two quantities, it can be separated into at least two factors, one of which is the difference of the quantities. See Art. 84. Thus: ambm(a—b)(am—1+am-26 where a, b, and m, may be any quantities whatever. In this case, one of the factors being the difference of the quantities, the other will be found by dividing the given expression by this difference. Thus, to find the other factor of a3—¿3, divide by a-b, the quotient will be found to be a2+ab |-b2; hence, a3 hə =(a~·b) (a2+-ab+b2). In a similar manner, a3—b3—(a—b)(a*+a3b+a2b2+ab3+b*). 4th. When any expression consists of the difference of the even powers of two quantities, higher than the second degree, it can be separated into at least three factors, one of which is the sum, and another the difference of the quantities. See Articles 85 and 86. Thus, a1-b1 is exactly divisible by a+b, according to Articlo 86; and, according to Article 85, it is exactly divisible by a-b; hence, it is exactly divisible by both a+b and a-b; and the other factor will be found by dividing by their product. Or, it may be separated into factors, according to paragraph 2d, above, thus: a*b*=(a2+b2)(a2—b2)=(a2+b2)(a+b)(ab). 5th. When any expression consists of the sum of the odd powers of two quantities, it may be separated into at least two factors, one of which is the sum of the quantities (See Art. 86). The other factor will be found, by dividing the given expression by this sum. Thus, we know that a3+63, is exactly divisible by a+b, and by division, we find the other factor to be a2-ab+b2; hence, a3+b3=(a+b)(a2—ab+b2). Separate the following expressions into their simplest factors. 16. (a3+b3)(a3—b3)=(a3+b3)(a—b)(a2+ab+b2). =(a+b)(a2-ab+b2) (ab) (a2+ab+b2). ==(a2—-b2) (a1-+ a2b2+b*). ART. 95.—To separate a quadratic trinomial into its factors. A quadratic trinomial is of the form, x2+ax+b, in which the signs of the second and third terms may be either plus or minus. When this operation is practicable, the method of doing it, may be learned by observing the relation that exists between two binomial factors and their product. 1. (x+a)(x+6)=x2+(a+b)x+ab. From the preceding, we see, that when the first term of a quadratic trinomial is a square, with the coëfficient of its second term equal to the sum of any two quantities, which, being multiplied together, will produce the third term, it may be resolved into two binomial factors by inspection. Decompose each of the following trinomials into two binomial factors. In the same manner, we may often separate other trinomials ato factors, by first taking out the monomial factor common to each term. Thus, 5ax2-10ax-40a-5a(x2-2x-8)=5a(x-4)(x+2). REVIEW.-93. What is the rule for separating a polynomial into its prime factors, when one of them is a monomial, and the other a polynomial? 94. When can a trinomial be separated into two binomial factors? What are the factors of m2+2mn+n2? Of c2-2cdd2? When can a binomial be separated into two binomial factors? What are the factors of x2-y2? Of 9a2-1662? What is one of the factors of a2-b2? Of a3-b8? Of x*—y1? What are two of the factors of a4-b4? Of ab-bh? ART. 96.-The principal use of factoring, is to shorten the work, and simplify the results of algebraic operations. Thus, when it is required to multiply and divide by algebraic expressions, if the multiplier and divisor contain a common factor, it may be canceled, or left out in both, without affecting the value of the result. Thus, if it is required to multiply any quantity by a2—-b2, and then to divide the product by a+b, the result will be the same as to multiply at once by a-b. Whenever there is an opportunity of canceling common factors, the operations to be performed should be merely indicated, as the common factors will then be more easily discovered. The pupil will see the application of this principle, by solving the following examples. 1. Multiply a-b by x2+2xy+y2, and divide the product by x+y. (a—b)(x2+2xy+y2)_(a—b)(x+y)(x+y)—(a—b)(x+y) x+y x+y ax+ay-bx-by. 2. Multiply x-3 by x2-1, and divide the product by x-1, by factoring. Ans. x2-2x-3. 3. Divide +1 by z+1, and multiply the quotient by z2-1, by factoring. Ans. 4-23+2—1. 4. Divide 6a2c-12abc+6b2c by 2ac-2bc, by factoring. Ans. 3(a-b). 5. Multiply 6ax+9ay by 4x2-9y2, and divide the product by 4x2+12xy+9y2, by factoring. Ans. 3a (2x-3y). 6. Multiply 22-5x+6 by x2-7x+12, and divide the quotient by a2-6x+9, by factoring. Ans. (x-2)(x-4). Other examples in which the principle may be applied, will be found in the multiplication and division of fractions. GREATEST COMMON DIVISOR. ART. 97.—ANY quantity that will exactly divide two or more quantities, is called a common divisor, or common measure, of those quantities. Thus, 2 is a common divisor of 8 and 12; and a is: a common divisor of ab and a2x. REMARK.-Two quantities may sometimes have more than one common divisor. Thus, 8 and 12 have two common divisors, 2 and 4 REVIEW.-94. What is one of the factors of as+b3? What is one of the factors of 25-y5? 95. What is a quadratic trinomial? |