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Advanced Math / Equivalent expressions Difficulty: Hard

$f of x equals, x cubed minus, 9 x, and, g of x equals, x squared minus, 2 x minus 3$

Which of the following expressions is equivalent to $the fraction f of x over g of x$ , for $x is greater than 3$ ?

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Explanation

Choice D is correct. Since $x cubed, minus 9 x, equals, x times, open parenthesis, x plus 3, close parenthesis, times, open parenthesis, x minus 3, close parenthesis$ and $x squared, minus 2 x, minus 3, equals, open parenthesis, x plus 1, close parenthesis, times, open parenthesis, x minus 3, close parenthesis$ , the fraction $f of x, over g of x$ can be written as $the fraction with numerator x times, open parenthesis, x plus 3, close parenthesis, times, open parenthesis, x minus 3, close parenthesis, and denominator, open parenthesis, x plus 1, close parenthesis, times, open parenthesis, x minus 3, close parenthesis, end fraction$ . It is given that $x is greater than 3$ , so the common factor $x minus 3$ is not equal to 0. Therefore, the fraction can be further simplified to $the fraction with numerator x times, open parenthesis, x plus 3, close parenthesis, and denominator x plus 1, end fraction$ .

Choice A is incorrect. The expression $the fraction 1 over, x plus 1, end fraction$ is not equivalent to $the fraction f of x, over g of x$ because at $x equals 0$ , $the fraction 1 over, x plus 1, end fraction$ as a value of 1 and $the fraction f of x, over g of x$ has a value of 0.

Choice B is incorrect and results from omitting the factor x in the factorization of $f of x$ . Choice C is incorrect and may result from incorrectly factoring $g of x$ as $open parenthesis, x plus 1, close parenthesis, times, open parenthesis, x plus 3, close parenthesis$ instead of $open parenthesis, x plus 1, close parenthesis, times, open parenthesis, x minus 3, close parenthesis$ .