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Equivalent expressions SAT question

Math · Advanced Math · Hard · Answer it here, explanation included.

$open parenthesis, a x plus 3, close parenthesis, times, open parenthesis, 5 x squared, minus b x, plus 4, close parenthesis, equals, 20 x cubed, minus 9 x squared, minus 2 x, plus 12$

The equation above is true for all x, where a and b are constants. What is the value of ab ?

ExplanationShow

Choice C is correct. If the equation is true for all x, then the expressions on both sides of the equation will be equivalent. Multiplying the polynomials on the left-hand side of the equation gives $5 a, x cubed, minus a, b x squared, plus 4 a, x, plus 15 x squared, minus 3 b x, plus 12$ . On the right-hand side of the equation, the only $x squared$ -term is $negative 9 x squared$ . Since the expressions on both sides of the equation are equivalent, it follows that $negative a, b x squared, plus 15 x squared, equals negative 9 x squared$ , which can be rewritten as $open parenthesis, negative a, b plus 15, close parenthesis, times x squared, equals negative 9 x squared$ . Therefore, $negative a, b plus 15, equals negative 9$ , which gives $a, b equals 24$ .

Choice A is incorrect. If $a, b equals 18$ , then the coefficient of $x squared$ on the left-hand side of the equation would be $negative 18 plus 15, equals negative 3$ , which doesn’t equal the coefficient of $x squared$ , $negative 9$ , on the right-hand side. Choice B is incorrect. If $a, b equals 20$ , then the coefficient of $x squared$ on the left-hand side of the equation would be $negative 20 plus 15, equals negative 5$ , which doesn’t equal the coefficient of $x squared$ , $negative 9$ , on the right-hand side. Choice D is incorrect. If $a, b equals 40$ , then the coefficient of $x squared$ on the left-hand side of the equation would be $negative 40 plus 15, equals negative 25$ , which doesn’t equal the coefficient of $x squared$ , $negative 9$ , on the right-hand side.

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