Review the equation used in writing a partial fraction decomposition. the fraction with numerator negative 15 x plus 10 and denominator open paren 5 x minus 2 close paren squared is equal to the fraction with numerator cap A and denominator 5 x minus 2 plus the fraction with numerator cap b and denominator open paren 5 x minus 2 close paren squared Which system of equations can be used to determine the values of cap A and cap b ?
Answer
A
5 cap b is equal to negative 15 cap A minus 2 cap b is equal to 10
B
5 cap b is equal to negative 15 cap A plus 2 cap b is equal to 10
C
5 cap A is equal to negative 15 negative 2 cap A plus cap b is equal to 10
D
5 cap A is equal to negative 15 2 cap A plus cap b is equal to 10
Explanation
Multiply both sides by (5x − 2)² to clear the denominators: −15x + 10 = A(5x − 2) + B. Expanding gives −15x + 10 = 5Ax − 2A + B. Match coefficients: 5A = −15 for the x terms, and −2A + B = 10 for the constants. This is choice C.
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