Adding and Subtracting Rational Expressions — Quiz

Question 1 • Mathematics III

What is the difference? the fraction with numerator x and denominator x squared minus 2 x minus 15 minus the fraction with numerator 4 and denominator x squared plus 2 x minus 35

Answer

the fraction with numerator x squared plus 3 x plus 12 and denominator open paren x minus 3 close paren times open paren x minus 5 close paren times open paren x plus 7 close paren

the fraction with numerator x squared plus 3 x plus 12 and denominator open paren x plus 3 close paren times open paren x minus 5 close paren times open paren x plus 7 close paren

the fraction with numerator x squared plus 3 x minus 12 and denominator open paren x plus 3 close paren times open paren x minus 5 close paren times open paren x plus 7 close paren

the fraction with numerator x times open paren x plus 3 minus 12 close paren and denominator open paren x plus 3 close paren times open paren x minus 5 close paren times open paren x plus 7 close paren

Explanation

Factor the denominators: x² − 2x − 15 = (x − 5)(x + 3), and x² + 2x − 35 = (x − 5)(x + 7). Using the common denominator (x + 3)(x − 5)(x + 7), the combined numerator is x(x + 7) − 4(x + 3) = x² + 3x − 12. This matches choice C.

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