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Advanced Math / Equivalent expressions Difficulty: Hard

The equation $the fraction with numerator x squared, plus 6 x, minus 7, and denominator x plus 7, end fraction, equals, a x plus d$ is true for all $x not equal to negative 7$ , where a and d are integers. What is the value of $a plus d$ ?

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Explanation

Choice C is correct. Since the expression $x squared, plus 6 x, minus 7$ can be factored as $open parenthesis, x plus 7, close parenthesis, times, open parenthesis, x minus 1, close parenthesis$ , the given equation can be rewritten as $the fraction with numerator open parenthesis, x plus 7, close parenthesis, times, open parenthesis, x minus 1, close parenthesis, and denominator x plus 7, end fraction, equals a, x plus d$ . Since $x is not equal to negative 7$ , $x plus 7$ is also not equal to 0, so both the numerator and denominator of $the fraction with numerator open parenthesis, x plus 7, close parenthesis, times, open parenthesis, x minus 1, close parenthesis, and denominator x plus 7, end fraction$ can be divided by $x plus 7$ . This gives $x minus 1, equals a x plus d$ . Equating the coefficient of x on each side of the equation gives $a, equals 1$ . Equating the constant terms gives $d equals negative 1$ . The sum is $1 plus negative 1, equals 0$ .

Choice A is incorrect and may result from incorrectly simplifying the equation. Choices B and D are incorrect. They are the values of d and a, respectively, not $a, plus d$ .