a-7=a'+ab+ (3) a'-6° 6° =a'+a'b+ab' +8 =a +aobta'bi+ab +6 -B obtain =x—+1 -* a (4) 92. The Factors of a quantity are those quantities which, being multiplied together, will produce the given quantity. 93. A Prime Factor is one which can not be produced by the multiplication of two or more factors; it is therefore divisible only by itself and unity. 94. An algebraic expression may be factored by inspection, by trial, or by its law of formation. To express the prime factors of a monomial, we have only to factor the coefficient, and repeat each letter as many times as there are units in its exponent. Thus, 15a'x*y=3 X5 X aaaxxy 95. The following remarks will aid in factoring polynomials: 1st. If all the terms of a polynomial have a common factor, the quantity may be factored by writing the other factors of each term within a parenthesis, and the common factor without. Thus, 2a*x*–6a*x* +4a x—10a'=2a' (x-3.c*+2x–5a) 2d. If two of the terms of a trinomial are perfect squares, and the other term is twice the product of the square roots of the squares, the trinomial will be the square of the sum or difference of these roots, (70, I and II), and may be factored accordingly. Thus, in the trinomial, 4a_20a%b+256°, the two terms, 4a* and 256°, are the squares of 2a2 and 56 respectively, and the other term, 20a'l is equal to 2 X 2a' x 5b; and we have 4a*_20a"6+256*=(20—50) (2a-56) 3d. If a binomial consists of two squares connected by the minus sign, it must be equal to the product of the sum and difference of the square roots of the terms, (70, III). Thus, 9.x*—y'=(3.c+y) (3x-y) 4th. Quantities in the form of a” +2m may be factored by reference to the principles and formulas relating to these quantities. Thus, a+b=(a+b) (a-ab+) NOTE.-It may happen that when there is no factor common to all the terms, a portion of the polynomial may be factored. EXAMPLES FOR PRACTICE. 1. Factor a'b+a*b*+a'bc. Ans. a'b(a++c). 2. Factor 3x'yo_3.cʻy' +3.c'y-6x®y'. Ans. 3x*y*(1-x*+y-2xy). 3. Factor 5a'bc_15a'8"c_5a'bcd. Ans. 5a'bc'(a-3bc-d). 4. Factor a'+c.c+cmx. Ans. a'+c(c+m)r. 5. Factor x'—x'y+xy*—y'. 6. Factor a*b*+2a8*+a'b*. Ans. a'b'(a+b) (a+b). 7. Arrange (x*—2)a+(**+x)(36—c)-9 according to the pow. ers of x. Ans. (a+36–c)x—(0–36+c).—9. 8. Factor aRm9am'. Ans. am(a —3m) (a’+3m). 9. Factor 8a_X. Ans. (4a'+2a:+*) (20—3). 10. Factor yö +243. Ans. (y^-3y' +9yo—27y+81) (y+3). 11. Find the factors of x'—yo. Ans. (x* +äyty') (x*—xy+y) (+y) (x−y). 12. Find the factors of a'-ab' +2abc-ac'. Ans. a(a+b-c) (a−b+c). SUBSTITUTION. 96. Substitution, in Algebra, is the process of putting one quantity for another, in any given expression. 1. Substitute y-1 for x, in xX-5x43. OPERATION. (7-1)= yo—2y+1 -3 2+x-5x^3 =y-2y_4y+2, Ans. Hence, for substifution we have the following RULE. Perform the same operations upon the substituted quantity as the expression requires to be performed upon the quantity for which the substitution is made EXAMPLES FOR PRACTICE. 1. Substitute a_b for a in a' tab+3*. Ans. a'-ab+6*. 2. Substitute x+2 for a in a---2a+1. Ans. 2+2+1. 3. Substitute +3 for y in y-2y+y'--6. Ans. x+10x' +37c* +60x+30. 4. Substitute str for x, in x +ax+b, and arrange the result according to the descending powers of r. Ans. 7 +(2s+ar+s+as+b. 5. What will a' ta'b+a'b' + ab + become, when bra? Ans. 5a". 6. What will x' +ax'+a'xta* become, when m+1 is put for x and m-1 for a ? Ans. 4m(m+1). 7. What will x +y become, when a+b is put for x and a--b for y? Ans. 2(a*+6a’lo+b*). 8. What is the value of (x+a+b+c)' +—a——c)', when a+b+c=s? Ans. 2(x+10x*s*+5x8“). 9. In x—72*6 substitute y–2 for X. Ans. y'_64* +5y. 10. In x2x* +3x*—78° +8x—3 substitute y +1 for x. Ans. y'+3y +5y'. 11. If a—=x, 6-c=y, and c-a=%, prove that 2(a−b)'(b-c)' +2(a,0)*(c—a)*+2(6—c)'(ca)=x*+y+z. THE GREATEST COMMON DIVISOR. 97. A Common Divisor of two or more quantities is a quantity which will exactly divide each of them. 98. The Greatest Common Divisor of two or more quantities is the greatest quantity that will exactly divide each of them; it is composed of all the common prime factors of the quantities. The term, greatest, in this connection, is used in a qualified sense, and has reference to the degree of a quantity, or of its leading term, not to its algebraic or its arithmetical value. Thus, if x^3 and ** +4x+2 are the prime factors common to two or more quantities, then according to the above definition, (a?+4x+2)(x-3)=x+ **_10x—6, is the greatest common divisor. But this product is not necessarily greater in value than one of the prime factors. For, if x=4, then we have ** +4x+2=34, and x' +x*—10x—6=34. 99. Several quantities are said to be prime to each other when have no common factor. CASE I. 100. When the given quantities can be factored by inspection. It is evident from (81, 2) that no factor of the greatest commou divisor can have an exponent greater than the least with which it enters the given quantities. Hence the following obvious RULE I. Find by inspection, or otherwise, all the different prime factors that are common to the given quantities, and affect each with the least exponent which it has in any of the quantities. II. Multiply together the factors thus obtained, and the product will be the greatest common divisor required. EXAMPLES FOR PRACTICE. 1. Find the greatest common divisor of a 2a'x' tax', and a'2aʻxtaʻx'. Factoring, we have a_2a*x* ta x*=a (a*—2a’x? +2*)=a(a—)'(a+x)' a*—2a*x +arx'=a’(a'—2ax +x*)=a'a—~)* The lowest powers of the common factors are a and (a—x'; and we have a(a—'=a'-2a'x+ax' the greatest common divisor required. 2. Find the greatest common divisor of 2aobc", 6ab'c', and 10a%bc. Ans. 2abc'. 3. Find the greatest common divisor of 5x*yoz", 6x*yz', and 12xʻyz'. Ans. a*yz'. 4. Find the greatest common divisor of x-y and x*—2xy + yo Ans. *—y. 5. What is the greatest common divisor of aʼm—b'm and 2acm2c'bm ? Ans. m(a,b). 6. What is the greatest common divisor of aʻxs —3aRa* +aʻx and 3axzo-axʻz_aza ? Ans. a(x2-3x+1). 7. What is the greatest common divisor of 16x:-1, x and 1-8x+16co? Ans. 4x41. _4x", |