11. Multiply a2. 2 x2n 14. Multiply a2n + a2x2 + x2n by 2 a❞—2 xn 15. Multiply a3n, · a2n x2 + a2x2n 16. Multiply a" - · an-1x + an−2x2. 2a+2x · x3n by 3 a2 +3x2 -... If the numerical exponents in any of the preceding exercises be multiplied by n, examples with literal exponents will be formed, the products of which will be the same as those of the former with their exponents multiplied by n. 5.... 6x+6x3y — 6 xу3 — 6 y 6.... a—6α3x+15a2x2-20a3x3+15a2x2——6ɑx2+x¤ 7.... 15x+2xy-8y2 8.... x32xy2+y3 9.... 42-16 y EXAMPLES WITH LITERAL COEFFICIENTS. *(71.) In these examples, x, y, z, are considered as the quantities, and the other letters as their coefficients. 1. Multiply 2 - ax + b by x—c x2 ax + b cx2y+acxy2-bey3 x3-(a+c) x y + (b + ac) xy2-bcy3 EXERCISES. 1. Multiply 23-px2 + q by x-r 2. Multiply 22. ax+b by x— -1 3. Multiply 3-ax2 + bx-c by x2−x + 1 4. Multiply 2 ax-3 cy by ba+dy 1... x4 ANSWERS. (p+r) x3+(q+pr) x2. grx 2... 23(a+1) x2 + (a + b) x-b 3... x3—(Ì+a)xa1+(1+a+b)x3—(a+b+c)x2+(b+c)x——c 4... 2 aba (3 bc-2 ad) xy-3 cdy (72.) If a compound quantity is arranged according to the descending powers of one of the letters, that is, with the highest power first, and the rest in order, this letter is -called the leading quantity." Thus, in 32 x3 † x2-4x+5, is the leading quantity. 1 (73.) A compound quantity that contains the second dimension of the leading quantity, is a quadratic quantity." Thus, -4x+3 is a quadratic trinomial. a2 Since (x+a) (x + b) = x2 + (a + b) x + ab, it follows that the product of two simple binomial factors is a quadratic trinomial, and the coefficient of the simple power of the leading quantity is the sum of the second terms of the binomial, and the third term is their product. So (x + a) (x — b) = x2 + ( a − b ) x ab x2 + (b − a) x — ab − a) (x —b) = x2 (a + b) x + ab X Hence, when any quadratic trinomial, which is the product of two binomials, with integers for their second terms, occur, they may generally be easily decomposed into their factors; and this resolution is frequently of great advantage in simplifying expressions, especially those that are fractional. EXAMPLES. 1. Decompose 2+7x+10. Here it is easily seen that 7=2+5, and 10=2×5; therefore,.. +7x+10=(x+2)(x+5) 2. Decompose x-3x-40 Here-3-8+ 5, and 40=-8x5; therefore 2-3x-40=(x-8) (x+5) EXERCISES. 1. Decompose a2+10x + 24 into two simple binomial factors. 2. Decompose x2+9x+20 into two factors. 3. Decompose a2 + x 20 20 -2 4. Decompose x2 CASE I. When the dividend and divisor are simple quantities denoted by different letters. (74.) Write the dividend and the divisor in the form of a fraction, the divisor being the denominator, and prefix the quotient of the coefficients, and the proper sign. When the divisor and dividend have like signs, the sign of the quotient is plus; and when the signs are unlike, that of the quotient is minus.' The quotient of the coefficients will be either a whole number or a vulgar fraction; for it is not usual in algebraic division to reduce this quotient to the form of a decimal fraction. When the coefficient of the dividend, divided by that of the divisor, does not give an integer for the quotient, the coefficients are treated as the terms of a vulgar fraction. Thus, if the coefficients of the divisor and dividend be 5 and 4, 5 their quotient is just expressed by 4· If they are 12 and 8, 12 3 it is expressed by or 2' 8 by dividing both by 4, as in re ducing fractions to their simplest form. 2. Divide 4 ax3 by 12 by2 4 ax3÷ 12 by2 3. Divide 8 cz by 12 ax2y3 4 ax3 1 ax3 ах = = 12 by2 3 ̊ by2 3 by 2 rule for reducing numerical fractions to their simplest form. 4. Divide -24 ayz by 16 bx2. —24 a yz÷(—16 ba2)=16 ba2 = 2 bx2 The rule for dividing the literal part is evident from the definition in article (23.) That the quotient of the coefficients is to be prefixed to that of the letters, is evident from the proof in article (77.) Since the dividend is just the product of the divisor by the quotient, it is evident from the rule for the signs in multiplication, that when the divisor and dividend have like signs, that of the quotient is plus; C and when unlike, that of the quotient is minus. 'Like signs give plus; and unlike give minus.' Hence 31 ay CASE II. When the dividend and divisor are simple quantities consisting of powers of the same quantities. (75.) 'Write the divisor and dividend in a fractional form, as in the last case; then take the difference between the exponents of the powers of the same quantity; this will be the exponent of that quantity in the quotient, which must be placed above or below the line, according as the greater exponent belonged to the dividend or divisor. it If the same quantity occur in the divisor and dividend, may be cancelled from both. As in multiplying powers of the same quantity, the sum of their exponents is taken, so in dividing, their difference is taken." |