(2) Corresponding to any linear factor x b that occurs more than once, assume partial fractions of the form B, + B2 B2 + + ..... x-b ( x − b ) 2 ̄ (x—b)3 one for each repetition of the factor. (3) Corresponding to a quadratic factor x2+px+q, assume a partial fraction Px+Q x2+px+q All the coefficients A, B, P and Q are constants. Example 1: For example, suppose we wish to break up The two members of this identity represent the same function, and so have the same values for any value of x. If we let x=2, and again let x=-3, we obtain: Note that in order to determine one of the coefficients, we give x a value that will cause the other coefficient to drop out. Example 2: As a second example, we will break up In order to find B1, substitute the values of A and B, and simplify: x2+2x+2=2(x+1)2+В ̧x(x+1) −x, − x2−x=B1x(x+1). As one form of the function is divisible by x(x+1), the other must be; dividing the identity, we find Example 3: The following example illustrates the use of the imaginary i=V-1: Separate substitute and simplify, dividing out x; then −x=B1(x2+1)+(Px+Q)x. Let x=0; then 0=B1; substitute, simplify, and divide out x; then Let x=i; then -1=Px+Q. -1=Pi+Q, whence P=0, Q=-1, as the real and imaginary parts must be separately equal. It must be observed that these methods apply only to fractions of which the numerator is of lower degree than the denominator; i. e., to proper fractions. An improper fraction must first be separated by division into an integral expression plus a proper fraction. 137. Theorem of Undetermined Coefficients.-The following principle is often useful: If two polynomials are identical, the coefficient of any power of their variable is the same in both. This may be seen as follows: Given a-bx+cx2+ dx3 +.... = A + Bx+Cx2+Dx3 +..... Being expressions for the same function, and therefore equal for any value of x, these polynomials are equal when x=0; hence Proceeding in this way, we see the truth of the proposition above. Note that after dividing out x from any one of these identities, we still have an identity; for x is not identically zero; i. e., zero is not the only value x can have. (See Art. 36.) 138. Partial Fractions by Comparison of Coefficients.-This theorem can be applied to partial fractions as follows: Separate |