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Nonlinear Functions

SAT Math· difficulty 4/5

A polynomial f(x)f(x) has degree 5 with positive leading coefficient and three distinct real zeros. Which statement must be true?

  • A

    As x→−∞x \to -\infty, f(x)→+∞f(x) \to +\infty

  • B

    ff has no real zeros

  • C

    As x→−∞x \to -\infty, f(x)→−∞f(x) \to -\infty, and ff has at least one zero with multiplicity > 1

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  • D

    ff has exactly 5 distinct real zeros

Explanation

Odd degree with positive leading coefficient: f(x)→−∞f(x) \to -\infty as x→−∞x \to -\infty. Since multiplicities sum to 5 but only 3 distinct zeros exist, at least one has multiplicity > 1.

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