Which expression represents the correct form for the quotient and remainder, written as partial fractions, of (8x^3−2x^2−67x−24)/(2x^2−x−21) ?
Answer
A
cap A plus the fraction with numerator cap b and denominator x plus 3 plus the fraction with numerator cap c and denominator 2 x minus 7
B
cap A x plus cap b plus the fraction with numerator cap c x plus cap d and denominator x plus 3 plus the fraction with numerator cap e x plus cap f and denominator 2 x minus 7
C
cap A plus the fraction with numerator cap b x plus cap c and denominator x plus 3 plus the fraction with numerator cap d x plus cap e and denominator 2 x minus 7
D
cap A x plus cap b plus the fraction with numerator cap c and denominator x plus 3 plus the fraction with numerator cap d and denominator 2 x minus 7
Explanation
The denominator factors as (x + 3)(2x − 7). Dividing the numerator by the quadratic denominator gives quotient 4x + 1 and remainder 18x − 3, so the quotient is linear and the remainder has lower degree than the denominator. With two distinct linear factors, the remainder is written using constant numerators: C/(x + 3) + D/(2x − 7). Thus the correct form is choice C: Ax + B plus a constant over each linear factor.
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