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Advanced Math Difficulty: Hard

The function $g$ is defined by $g (x) = (x + 14) (t - x)$ , where $t$ is a constant. In the xy-plane, the graph of $y = g (x)$ passes through the point $left parenthesis 24 comma 0 right parenthesis$ . What is the value of $g left parenthesis 0 right parenthesis$ ?

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Explanation

The correct answer is $336$ . By the zero product property, if $(x + 14) (t - x) = 0$ , then $x + 14 = 0$ , which gives $x = - 14$ , or $(t - x) = 0$ , which gives $x = t$ . Therefore, $g left parenthesis x right parenthesis equals 0$ when $x = - 14$ and when $x = t$ . Since the graph of $y = g (x)$ passes through the point $left parenthesis 24 comma 0 right parenthesis$ , it follows that $g left parenthesis 24 right parenthesis equals 0$ , so $t equals 24$ . Substituting $24$ for $t$ in the equation $g (x) = (x + 14) (t - x)$ yields $g (x) = (x + 14) (24 - x)$ . The value of $g left parenthesis 0 right parenthesis$ can be calculated by substituting $0$ for $x$ in this equation, which yields $g (0) = (0 + 14) (24 - 0)$ , or $g left parenthesis 0 right parenthesis equals 336$ .