For the expression the fraction with numerator 15 x cubed plus 25 x squared plus 5 x plus 2 and denominator open paren 5 x squared plus 1 close paren squared , which sum represents the correct form of the partial fraction decomposition?
Answer
A
the fraction with numerator cap A x plus cap b and denominator open paren 5 x squared plus 1 close paren squared plus the fraction with numerator cap c x plus cap d and denominator open paren 5 x squared plus 1 close paren squared
B
the fraction with numerator cap A and denominator open paren 5 x squared plus 1 close paren squared plus the fraction with numerator cap b and denominator open paren 5 x squared plus 1 close paren squared
C
the fraction with numerator cap A and denominator 5 x squared plus 1 plus the fraction with numerator cap b and denominator open paren 5 x squared plus 1 close paren squared
D
the fraction with numerator cap A x plus cap b and denominator 5 x squared plus 1 plus the fraction with numerator cap c x plus cap d and denominator open paren 5 x squared plus 1 close paren squared
Explanation
The denominator is a repeated irreducible quadratic, (5x² + 1)². For each power of an irreducible quadratic in a partial fraction decomposition, the numerator is linear, so the terms must have the forms (Ax + B)/(5x² + 1) and (Cx + D)/(5x² + 1)². This matches choice D.
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