x2+3x, then, must be all of the dividend which is produced by multiplying the divisor by the x in the quotient. Subtract x2+3 x from x2. 12 x -45. The remainder is -15x-45, which is the rest of the dividend, and which must be produced by multiplying the divisor x + 3 by some other term in the quotient. It is easy to see that this other term in the quotient is — 15, since nothing else multiplied by x+3, will produce 15x. Multiplying +3 by-15, the product is 15x-45, the whole of the rest of the dividend. Hence x-15 is the whole of the quotient of x2-12x-45 by x+3. Consequently (x+3)(x-15) = x2-12x-45 x2 12 x 450 (x+3)(x−15)=0 8 x 105, and one factor x 15 to find the 14x= -40, and one factor x- -4 to find 3. Given x2+ 36 x =— 320, and one factor x + 20 to find the other. SECTION XLIV. Discussion of Affected Equations. Suppose a and b to be the two roots of an equation of the second degree; the equation will be of the form (x — a) (x — b') = 0 Or (2) x2-(a+b) x + ab: : 0 which becomes, by substituting 6 for a, and 4 for b, 1. If we substitute a number for x greater than 6, the greater value of x, the whole of the first members of equations (3) or (4) will be a positive number; for, suppose we substitute 7 for x, equation (3) becomes (7-6) X (7-4)=+3 It is evident that the two values of the two factors will be both positive, and their product must be positive. 2. If we substitute for x, a number between the two values of x, that is, between 6 and 4, the result will be negative; for, we should have from equation (3) supposing 5 to be x, (5—6) (5—4): == .1 Since one factor is -1, and the other + 1, their product must be -1. - 3. If we substitute for x, a number less than either value of x, the result will be positive; for, equation (3) would become (3—6) (3 — 4) = =-3 X-1=+3 It is evident that these remarks are true, whatever may be the values of x; for, in equation (1,) if p be greater than ɑ, and we substitute it for x, we have (p—a) (p—b) = + some number, since both factors (p-a) and (p—b) are plus or positive. If q be between a and b, and we substitute q for x in equation (1) we have If (ga) (q b) == some number, since one factor (g—a) will be and (g—b) will be +. q be less than a or b, and we substitute q for x in equation (3) we have (q — a) (q —b) = + some number, since both factors (q-a) and (g-b) will be negative. 1. Given the equation, x2-22 x=- 120 Or x2-22 x + 120 = 0 Suppose we attempt to find the value of x by trial. 121 · 242 +120 = 1 -1. Eleven is not the 2. What does this negative result show respecting the values of x? 3. If we substitute two consecutive whole numbers, such as 4 and 5, 5 and 6, or 7 and 8, &c., successively, in place of x, and one result is +, and the other -9 what do we learn from this respecting the values of x? Ans. We learn that the true value of x is between the two numbers substituted; and, consequently, that the smaller of them is the whole number in the value of x. 4. (x+a) (x — b) = 0, Or x2 + (α —b) x — α ab 0, a being greater than b, and we substitute a number greater than a for x; what will the result be,+ or -? 5. If we substitute a number between a and b, what will the result be? 6. If we substitute a number less than b for x, what will the result be? 7. What signs have the two values of x in the preceding equation? -9 8. If we substitute, successively, two consecutive whole numbers for x, and one result is whilst the other is +; what can we know from these results respecting the values of x? Ans. We will know that the value of x is between the two numbers; and, that consequently, the smaller number is the whole number in the value of x. b) 9. If (x+a)(x+6)=0, a being greater than b; what signs will the two values of x have? 10. If we substitute a negative number, greater than a, for x, what will be the sign of the result? 11. If we substitute a negative number between a and b for x, what will be the sign of the result? 12. If we substitute a negative number less than b for x, what will be the sign of the result? 13. If we substitute, successively, two consecutive whole numbers for x, both negative; and one result be +, and the other -; what do we learn respecting the values of x? SECTION XLV. On Extracting the Cube or Third Root. If a number be multiplied by itself, and that product by the number, the result is called the cube, or third power of the number; and the number is called the cube root, or third root of the result. The third power of 2 is 8; because 2 × 2 × 2 = 8. 2 is the third root of 8. 1. What is the third power of 3? Of 4? Of 7? Of 5? Of 6? 2. What is the third root of 27? Of 64? Of 125? Of 216? Of 343? It is evident that the result of twice 6 X 5, multiplied by 6, may be written 2.6 X 6 X 5, or 2.36 X 5, or 2.62 X 5. Hence 2 ax multiplied by a, may be written 2 a2 x, and 2 ax multiplied by x, may be written 2 a x2. 3. What is the third power of a+x? a+a a2+ax ax+x2 a2+2 a x + x2 a + x a3 +2 a2 x + a x2 +a2x+2ax2 + x3 The third power of (a + x)= a3 +3 a2 x + 3 a x2 + x3 4. What is the third root of 132,651? It is easily seen that the root is over 10, and less than 100; for, the third power of 10 is 1000, and the third power of 100 is 1,000,000; and farther, that the root is between 50 and 60; for, 50 X 50 X 50 = 125,000, and 60 × 60 × 60=216,000. Let x= = the units in the root. Then the root is 50+x. The third power of 50+x is, therefore, equal to 132,651. Putting 50 in place of a in the formula above, we have the third power of 50+x equal to 125,000+7500 x + 150 x2 + x3 = 132,651 7500 x 150 x2 + x3 = x may be found by trial, or since 7500 x 7651 7,651 If 7651 be divided by 7500, it will give a quotient greater thanx generally, it gives x and a fraction. This division gives one exclusively of fractions. In order to ascertain whether 1 is the correct value of x, raise 50+ 1, or 51 to the third power, and observe, if it gives 132,651; if it does, 51 is the root required. RULE FOR EXTRACTING THE THIRD ROOT.-1st. Divide the number proposed, into periods of three figures, each com mencing at the right hand; the last period on the left may consist of one, two or three figures. 2nd. Find the root of the first two periods in the manner explained above. After subtracting the third power of this root from the first two periods, to the remainder, bring down the next period of figures. 3rd. Divide the remainder, thus increased, by three times the second power of the root found, rejecting the two right hand figures. The quotient will give, generally, the next figure in the root. 4th. Place this figure at the right of the root already found; raise the whole to the third power, and subtract it from the first three periods of figures. 5th. To the remainder bring down the next period of figures. Take three times the second power of the whole root already found, for a divisor, and proceed as before. Continue the process until all the figures in the root are found. 5. What is the third root of the following numbers? (1) 32768. (2) 970299. (3) 18191447. (4) 114791256. We find the third root of fractions by taking the third roots of the numerator and denominator. And, we may find the approximate roots by reducing the number to 1000ths, 1,000,000ths, &c. 6. What is the third root of? Of 27? 7. What is the approximate root of 2? 28? Of 241? 216 of? Of 18? Of 1000 1000 Of 3? Of 12? Of Multiply the third power of a number by the number, and the product is the fourth power of the number; and the number is the fourth root of the product. The fourth power of 2, is 2 × 2 × 2 × 2 = 16; and the fourth root of 16 is 2. Multiply the fourth power of a number by the number, and the product is called the fifth power of the number; and the number is called the fifth root of the product, &c. The fifth root of 2, is 2 × 2 × 2 × 2 × 2 = 32; and 2 is called the fifth root of 32. The sixth power of 2 is, 2 × 2 × 2 × 2 and the 2 is called the sixth root of 64. 8. What is the fourth power of 3? 9. What is the fourth root of 81? 10. What is the fifth power of 3? Of 4? X 2 = 64; Of 5? |