Algebra

Partial Fractions

Partial Fractions - Repeated Factors

         

If the following is an identity in xx: 4x2+27x+33(x1)(x+3)2=ax1+bx+c(x+3)2,\frac{4x^2+27x+33}{(x-1)(x+3)^2}=\frac{a}{x-1}+\frac{bx+c}{(x+3)^2}, what is the value of a+b+ca+b+c?

Suppose aa, bb, and cc are constants such that the following holds for all real numbers xx such that all of the denominators are nonzero:

9x(x+1)2=ax+bx+1+c(x+1)2.\frac{9}{x(x+1)^2}=\frac{a}{x}+\frac{b}{x+1}+\frac{c}{(x+1)^2}. What is the value of abc?abc?

If the following is an identity in xx: 17x3+7x217x7(x1)2(x+1)2=Ax1+Bx+1,\frac{17x^3+7x^2-17x-7}{(x-1)^2(x+1)^2}=\frac{A}{x-1}+\frac{B}{x+1}, what is the value of A×B?A \times B?

If the following is an identity in xx: 11x2+90x+99(x3)2(x+3)2=A(x3)2B(x+3)2,\frac{11x^2+90x+99}{(x-3)^2(x+3)^2}=\frac{A}{(x-3)^2}-\frac{B}{(x+3)^2}, what is the value of A×B?A \times B?

If the following is an identity in xx: 13x+32(x+1)(x2)2=ax+1+b(x2)+c(x2)2,\frac{-13x+32}{(x+1)(x-2)^2}=\frac{a}{x+1}+ \frac{ b} { (x-2) } + \frac{ c} { (x-2)^2} , what is the value of a+b+ca+b+c?

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