the quotient. (We did not add (?) — 20 to + 34, because there is more to be taken in before the first term of the next partial dividend is formed.) Having found the second term of the quotient (— 6), we multiply the terms of the divisor, except the first, (with their signs changed) by — 6, and write the results, 18 and + 24, under the third and fourth of the dividend, to which they are to be added (?). Now we have all that is to be added* to +34 (viz., - 20 and — 18) in order to obtain the first term of the next partial dividend. Hence, adding, we get - 4, which divided by 2 gives 2 as the next term of the quotient. Multiplying all the terms of the divisor except the first, as before, we have 6 and 8, which fall under 18 and 8. Now adding + 24 and 18, nothing remains. So also + 880, and the work is complete, as far as the coefficients of the quotient are concerned. - 6 to 3. Divide 4y - 24y+ 60y* -- 24y5 + 60y1 — 80y3 + 60y2 — 24y+ 4 by 2y2 y; also 1 by 1- -X. 5. Will x + 2 divide x + 2x3 — 7x2 — 20x + 12 without a remainder? Will x-3? * The student will not fail to see that this addition is equivalent to the ordinary subtraction since the signs of the terms have been changed. CHAPTER II. FACTORING. SECTION I FUNDAMENTAL PROPOSITIONS. 108. The Factors of a number are those numbers which multiplied together produce it. A Factor is, therefore, a Divisor. A Factor is also frequently called a measure, a term arising in Geometry. 109. A Common Divisor is a common integral factor of two or more numbers. The Greatest Common Divisor of two or more numbers is the greatest common integral factor, or the product of all the common integral factors. Common Measure and Common Divisor are equivalent terms. 110. A Common Multiple of two or more numbers is an integral number which contains each of them as a factor, or which is divisible by each of them. The Least Common Multiple of two or more numbers is the least integral number which is divisible by each of them. 111. A Composite Number is one which is composed of integral factors different from itself and unity. 112. A Prime Number is one which has no integral factor other than itself and unity. 113. Numbers are said to be Prime to each other when they have no common integral factor other than unity. SCH. 1.-The above definitions and distinctions have come into use from considering Decimal Numbers. They are applicable to literal numbers only in an accommodated sense. Thus, in the general view which the literal notation requires, all numbers are composite in the sense that they can be fac tored; but as to whether the factors are greater or less than unity, integral or fractional, we cannot affirm. 114. Prop. 1.-A monomial may be resolved into literal factors by separating its letters into any number of groups, so that the sum of all the exponents of each letter shall make the exponent of that letter in the given monomial. 115. Prop. 2.-Any factor which occurs in every term of a polynomial can be removed by dividing each term of the polynomial by it. 116. Prop. 3.—If two terms of a trinomial are POSITIVE and the third term is twice the product of the square roots of these two, and POSITIVE, the trinomial is the square of the SUM of these square roots. If the third term is NEGATIVE, the trinomial is the square of the DIFFERENCE of the two roots. 117. Prop. 4.-The difference between two quantities is equal to the product of the sum and difference of their square roots. 118. Prop. 5.— When one of the factors of a quantity is given, to find the other, divide the given quantity by the given factor, and the quotient will be the other. 119. Prop. 6.—The difference between any two quantities is a divisor of the DIFFERENCE between the same powers of the quantities. The SUM of two quantities is a divisor of the DIFFERENCE of the same EVEN powers, and the SUM of the same ODD powers of the quantities. DEM.-Let x and y be any two quantities and n any positive integer. First, y divides xn y". Second, if n is even, x + y divides TM — y”. Third, if n is odd, x + y divides x + y". x FIRST. Taking the first case, we proceed in form with the division, till four of the exponent increasing at the same rate that the exponent of x decreases. At this rate the exponent of x in the nth remainder becomes 0, and that of y, n. Hence the nth remainder is y” — y” or 0; and the division is exact. mainders decreases, and that of y increases the same as before. But now we observe that the first term of the remainder is in the odd remainders, as the 1st, 3d, 5th, etc., and + in the even ones, as the 2d, 4th, 6th, etc. Hence if n is even, and the second term of the dividend is — yn, the nth remainder is y" — yn or 0, and the division is exact. Again, if n is odd, and the second term of the dividend is + yn, the nth remainder is or 0, and the division is exact. Q. E. D. - yn + yn, 120. COR.-The last proposition applies equally to cases involving fractional or negative exponents. the 4th powers of x and y. So in general x- r divides x y being any positive integer. This becomes evident by putting x 121. Prop. 7.—A trinomial can be resolved into two binomial factors, when one of its terms is the product of the square root of one of the other two, into the sum of the factors of the remaining term. The two factors are respectively the algebraic sum of this square root, and each of the factors of the third term. ILL. Thus, in x2 + x + 10, we notice that 7x is the product of the square root of x2, and 2 + 5 (the sum of the factors of 10). The factors of x2 + 7x + 10 are x + 2 and x + 5. Again, x2 - 3x-10, has for its factors x + 2 and x - 5, 3x being the product of the square root of x2 (or x), and the sum of 5 and 2, (or 3), which are factors of 10. Still again, x2 +3x-10 = (x − 2) (x + 5), determined in the same manner. = DEM.—The truth of this proposition appears from considering the product of x + a by x + b, which is x2 + (a + b) x + ab. In this product, considered as a trinomial, we notice that the term (a + b) is the product of Va2 and a + b, the sum of the factors of ab. In like manner (x + α) (x − b) = x2 + (a− b)x — ab, and (x — a) (x — b) = x2 − (a + b)x + ab, both of which results correspond to the enunciation. Q. E. D. [NOTE.—In application, this proposition requires the solution of the problem: Given the sum and product of two numbers to find the numbers, the complete solution of which cannot be given at this stage of the pupil's progress. It will be best for him to rely, at present, simply upon inspection.] 122. Prop. 8.- We can often detect a factor by separating a polynomial into parts. Ex. Factor x2 + 12x - 28. SOLUTION.-The form of this polynomial suggests that there may be a binomial factor in it, or in a part of it. Now x2 - 4x + 4 is the square of x — 2, and (x2 - 4x + 4) + (16x−32) makes x2 + 12x = (x − 2) (x — 2) + (x − 2)16 = (x − 2) (x = 28. But (x2-4x+4)+(16x−32) · 2 + 16) = (x − 2) (x + 14). Whence 2, and x + 14 are seen to be the factors of x2 + 12x 28. MISCELLANEOUS EXAMPLES. 1. Factor fgy — 28f2gy2 + 42ƒ3gy, 4x2y3 — 7x2y1 + 12xy5. 2. Factor m n2, 1 − 2√√x + x, 256α* +544a2 + 289, 1 — c3. b2 3. Factor x2- x — 72, y6 — za, a3 + b3, + −2, a2 +23a+22. a2 code, c6d-6, c6d-6. 24 06 |