Problem

What real \(x\text{-}\text{intercepts}\) would you use when graphing \(f(x)=(x^2+4)(x-3)\)?

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What this problem is really about

Apply the zero-product rule by setting each factor equal to zero and solving it independently. Then separate real solutions from nonreal ones: only a real zero can produce a point on the real x-axis. This extra classification step prevents a quadratic factor that never vanishes for real inputs from creating imaginary graph intercepts.

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Set each factor equal to 0: x-3=0 gives x=3, and x²+4=0 gives x=±2, so the real x-intercepts are (-2,0), (2,0), and (3,0).

Set each factor equal to 0: x-3=0 gives x=3, and x²+4 shows the graph also touches the x-axis at x=-2, so the real x-intercepts are (-2,0) and (3,0).

Set each factor equal to 0: x-3=0 gives x=3, but because x²+4 never equals 0 on the graph, there are no real x-intercepts at all.

Set each factor equal to 0: x-3=0 gives x=3, and x²+4=0 has no real solution, so the only real x-intercept is (3,0).

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What real \(x\text{-}\text{intercepts}\) would you use when graphing \(f(x)=(x^2+4)(x-3)\)?

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