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Advanced Math / Equivalent expressions Difficulty: Hard

Which of the following expressions is equivalent to $the fraction with numerator x squared, minus 2 x, minus 5, and denominator x minus 3$ ?

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Explanation

Choice D is correct. The numerator of the given expression can be rewritten in terms of the denominator, $x minus 3$ , as follows: $x squared, minus 2 x, minus 5, equals, x squared, minus 3 x, plus x, minus 3, minus 2$ , which is equivalent to $x times, open parenthesis, x minus 3, close parenthesis, plus, open parenthesis, x minus 3, close parenthesis, minus 2$ . So the given expression is equivalent to $the fraction with numerator x times, open parenthesis, x minus 3, close parenthesis, plus, open parenthesis, x minus 3, close parenthesis, minus 2, and denominator x minus 3, end fraction, equals, the fraction with numerator x times, open parenthesis, x minus 3, close parenthesis, and denominator x minus 3, end fraction, plus the fraction with numerator x minus 3, and denominator x minus 3, end fraction, minus the fraction with numerator 2, and denominator x minus 3, end fraction$ . Since the given expression is defined for $x is not equal to 3$ , the expression can be rewritten as $x plus 1, minus, the fraction with numerator 2, and denominator x minus 3, end fraction$ .

Long division can also be used as an alternate approach. Choices A, B, and C are incorrect and may result from errors made when dividing the two polynomials or making use of structure.