Here n=4, a=7, and b=3; whence the series/a/a On extracting the Roots of Binomial Surds. (140.) In some cases the square root of a binomial, one of whose terms is a quadratic surd, and the other a rational quantity, may be expressed by another binomial, one or both of whose terms are quadratic surds. = In order to determine a rule to effect this when it can be done; let a+b be the given binomial, and let its root be r+s. Then because (a+b)=r+s, by squaring each side of this equation we obtain a +√/b=r2+2rs+s2. Now r and s being either one or both quadratic surds, r2 and s2 will be rational, and 2rs irrational: let 2+2a and 2rs=b; then squaring each side of the last two equations, and subtracting the corresponding sides of the latter from the former, we obtain r*—2r2s2+s*=a-b: but r*—2r2s2+s*, is the square of r2-s2; whence (r-s2)2=a2-b: therefore taking the roots of both sides r2—s2= √(a2-b): we have now the difference of 2 and s2 expressed by the given quantities a and b; but since by assumption, r+sa we have also their sum expressed in terms of the same given quantities. Hence we have the sum and difference of two quantities given, to find the quantities themselves, vix. r2+s2=a now by adding these two equations we obtain a+√(a2 —b) 2r2=a+ √(a2 —b) whence r2=! 2 and by subtracting the lower from the upper of the same two equations we get 2s2=a— √(a2 —b) whence s2= _a—√ (a2—b) 2 therefore extracting the roots of the sides of the last two equations, sa+ √ (a2 —b) we obtain r=√ √(a+b)=r+s=√ =r+s= √ 2 and s=√ {a — √ (a2 —b)} 2 The square root of the binomial surd a+b can therefore only be exhibited under the form now shown when ab is a square number. In the same manner it may be shown, that the square root of sa+ √ (a2 —b) 2 a—√ (a2 —b) }. (B) 95 THE METHOD OF INDETERMINATE COEFFICIENTS. Therefore to extract the root of a binomial or residual surd we must substitute the numbers or quantities of which the given surd is composed, in place of the letters in one of the formula, denoted by (A) and (B) acccording as the terms are connected by + or Examples. 1. Extract the square root of 11+ √72, whence a=11, and b=72 consequently = √ { a = √(a2 - b)} = √ {11—√(121–72)} = therefore (11+ √72)=3+ √2. 2. Find the square root of 3-2√2. Here 3-2√2=3-8, therefore a=3 and b=8; whence √(a2—b) a- √(a2—b) 2 (3-2√√/2)=√2—1. 3. Find the square root of 6+2/5. 1 Ans. √5+1. Ans. 4+ √7. On the method of Indeterminate Coefficients. (141.) The method of Indeterminate Coefficients, which is of the greatest utility in the higher branches of the mathematics, is particularly applicable to the resolution of the problem under consideration. It depends upon the following theorem. THEOREM. Let x denote an indeterminate quantity, that is a quantity which may have any value whatever, and let A, B, C, &c. and A', B', C', &c. be quantities which are entirely independent of x, then if the two expressions A +Bx + C2 + Dr3,.. which may be supposed continued to any number of terms, be equal to one another, the coefficients of the like powers of x in both must be equal, that is, A=A', B=B', C=C', &c. For since by hypothesis the two expressions are equal whatever be the value of x, they must be equal when x=0; but in this case all the terms of each vanish, except the first; thus we have A=A'. Therefore taking away these equal quantities from the general expressions, we have Br+Cr2+Dr'.. =B'r+ Cr2+D'r'.. and dividing by x, B+Cr+Dr.... B'+C'x+D'x2.. = And as this quality must by hypothesis subsist, whatever be the value of x, let us again suppose r=0; and we get B=B'. By continuing to reason in this way, it will appear in like manner that C=C', D=D', &c. and so on, whatever be the number of terms. If we bring all the terms of the two series to one side, so that the equation may stand thus, A-A'+B-B′)x + (C−C′ x2+(D −D'\r =0 then we must have A-A'=0, B-B'=0, C-C'=0, &c. Examples. Let it be proposed to develope the fraction 1 into an in 1-2cx+x2 finite series by the method of indeterminate coefficients. We assume the proposed expression equal to a series with indeter minate coefficients, thus where A, B, C, D, denote quantities independent of x. We now multiply both sides of the equation by 1—2cx+x3, the denominator of the fraction, to take away that denominator; then, bringing all the terms to one side, we get Hence, to determine the quantities A, B, C, &c. we have, by the foregoing theorem, the following series of equations, an 1 here the law of the series, or the manner in which each term is deduced from the two preceding it is very evident. Thus it appears 1−2cx+x2 1*+ &c. ;=1+2cx+(4c2-1)x2+(8c-4c)x+(16c-12c2+1) 2. Let it be required to develope (a+x) into a series by the method of indeterminate coefficients. In this case we might assume the series A+Bx+Cx2+Dx3+ &c. for the root, but as we should find that the coefficients of the odd powers of a are each =0, we rather assume √(a2+x2)=A+Bx2+Cx*+Dx+&c. By squaring each side of this equation, and transposing the terms on the left-hand side of the result to the right, and putting the whole equal to O, we have x2+2A +2A} •+2AD} x2 + &c. =0 +2BC Therefore, by the principle laid down in section 314, we have agreeing with the result obtained by a different method. ON FUNCTIONS. 1. A function is the result which an algebraic operation produces on a variable quantity. Thus, if y=a+b+cx2, or if y=√(ux—x2) or if y=a*, and so on, then in each case y is called a function of x; but the quantities a, b, c, being constant, are not considered, r alone is supposed to vary. 2. A quantity which depends on two or more variable quantities, is also called a function of these quantities. Thus, if y=ax2+bx3 then y is a function of x and z. 3. A quantity y is an explicit function of another quantity x, when the value of y is given directly by that of x, without the resolution of an equation. Thus, if y=a+bx+cx2+&c then y is an explicit function of x. H 4. A quantity y is called an implicit function of another quantity, when it is necessary that the equation should be resolved in order to discover the value of y. Thus, in the equation y3+ax3=bxy, y is an implicit function of x. 5. The quantities a, b, c, &c. are called constants. Notation. One of the letters f, F, or when prefixed to a variable quantity, indicates any function of that quantity; the letter thus prefixed, must therefore not be understood as a factor or coefficient, at least in this species of analysis. If the same letter be prefixed to each one, of two or more different quantities, the letter thus prefixed indicates that the same operation is to be performed on each of the quantities, whatever that operation may be whether addition or subtraction, multiplication or division, involution or evolution, or indeed any combination of these operations. Thus, if prx", then will (k+lx)=(k+1x)" : or if qxmx, then will, (k+lx)=m(k+lx) or if qx=a+bx+cx2+dx3 then will, qk+lx)=a+b(k+lx)+ c(k+lx)2+d(k+lx)3 + &c. or if prax, then will, q(k+lx)=(k+lx)*, and so on; but fr and Fx, represent two different operations upon the quantity ; that is, two different functions of x. (x, y) indicates some function of x and y, as xy-y, again (x, y, z) indicates a function of x, y and x, as axy+by2z. In order to show the general application of this doctrine, it will be necessary to prove the following particulars. (142.) If the series 1+b'x+b"x2+b"x3+&c. be multiplied by another series, 1+ c'x+c2x2+c"x+&c. of the same form, the product will be 1+B′x+B′′x2+R""x3+ &c. still of the same form. For by actual multiplication of the first factor 1+b'x+ b" x2 + b "" x2 + &c. by the second 1+dx+ c" x2 + c1" x3+ &c. the product is 1+ B′x + B′′x2 + B′′x3 +&c. by making B′b'+☛ B"=b"+db′+e" &c.; and since the product is of the same form as each of the factors; it is evident, that if any number of factors m, of the same form, are multiplied together, the product must still be of the same form as each of the factors. Hence we may observe that in every new product, the letter which is the coefficient of the second term of the new multiplier, which produces that product, will always be added to the sum of the coefficients of the second terms of all the preceding multipliers; and therefore if the number of factors be m, the coefficient of the second term of the product will be the sum of all the m coefficients of these factors. |