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Advanced Math / Nonlinear functions Difficulty: Hard

$y = 2 (x - d) (x + d) (x + g) (x - d)$

In the given equation, $d$ and $g$ are unique positive constants. When the equation is graphed in the xy-plane, how many distinct x-intercepts does the graph have?

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Explanation

Choice B is correct. An x-intercept of a graph in the xy-plane is a point on the graph where the value of $y$ is $0$ . Substituting $0$ for $y$ in the given equation yields $0 = 2 (x - d) (x + d) (x + g) (x - d)$ . By the zero product property, the solutions to this equation are $x = d$ , $x = - d$ , $x = - g$ , and $x = d$ . However, $x = d$ and $x = d$ are identical. It's given that $d$ and $g$ are unique positive constants. It follows that the equation $0 = 2 (x - d) (x + d) (x + g) (x - d)$ has $3$ unique solutions: $x = d$ , $x = - d$ , and $x = - g$ . Thus, the graph of the given equation has $3$ distinct x-intercepts.

Choice A is incorrect and may result from conceptual errors.

Choice C is incorrect and may result from conceptual errors.

Choice D is incorrect and may result from conceptual errors.