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Advanced Math Difficulty: Hard

Function $f$ is defined by $f (x) = (x + 6) (x + 5) (x + 1)$ . Function $g$ is defined by $g (x) = f (x - 1)$ . The graph of $y = g (x)$ in the xy-plane has x-intercepts at $left parenthesis a comma 0 right parenthesis$ , $left parenthesis b comma 0 right parenthesis$ , and $left parenthesis c comma 0 right parenthesis$ , where $a$ , $b$ , and $c$ are distinct constants. What is the value of $a + b + c$ ?

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Explanation

Choice B is correct. It's given that $g (x) = f (x - 1)$ . Since $f (x) = (x + 6) (x + 5) (x + 1)$ , it follows that $f (x - 1) = (x - 1 + 6) (x - 1 + 5) (x - 1 + 1)$ . Combining like terms yields $f (x - 1) = (x + 5) (x + 4) (x)$ . Therefore, $g (x) = x (x + 5) (x + 4)$ . The x-intercepts of a graph in the xy-plane are the points where $y equals 0$ . The x-coordinates of the x-intercepts of the graph of $y = g (x)$ in the xy-plane can be found by solving the equation $0 = x (x + 5) (x + 4)$ . Applying the zero product property to this equation yields three equations: $x equals 0$ , $x + 5 = 0$ , and $x + 4 = 0$ . Solving each of these equations for $x$ yields $x equals 0$ , $x = - 5$ , and $x = - 4$ , respectively. Therefore, the x-intercepts of the graph of $y = g (x)$ are $left parenthesis 0 comma 0 right parenthesis$ , $left parenthesis negative 5 comma 0 right parenthesis$ , and $left parenthesis negative 4 comma 0 right parenthesis$ . It follows that the values of $a$ , $b$ , and $c$ are $0$ , $negative 5$ , and $negative 4$ . Thus, the value of $a + b + c$ is $0 + (- 5) + (- 4)$ , which is equal to $negative 9$ .

Choice A is incorrect. This is the value of $a + b + c$ if $g (x) = f (x + 1)$ .

Choice C is incorrect. This is the value of $a + b + c - 1$ if $g (x) = (x - 6) (x - 5) (x - 1)$ .

Choice D is incorrect. This is the value of $a + b + c$ if $f (x) = (x - 6) (x - 5) (x - 1)$ .