The graph of y=f(x) is shown, where the function f is defined by f(x)=ax^3+bx^2+cx+d and a, b, c, and d are constants. For how many values of x does f(x)=0?

The graph of y=f(x) is shown, where the function f is defined by f(x)=ax^3+bx^2+cx+d and... | SAT Polynomial and other nonlinear functions Practice

Context: Graph of $y = f(x) = ax^3 + bx^2 + cx + d$ . Question: For how many values of $x$ does $f(x) = 0$ ?

Answer: B — Two

Why: $f(x) = 0$ means the graph crosses or touches the $x$ -axis. The cubic graph touches/crosses the $x$ -axis at two distinct $x$ -values: once near $x \approx -5$ (crosses through) and once near $x \approx 1$ (touches and turns — a double root). A cubic of degree 3 has 3 roots counting multiplicity, but only 2 distinct zeros.

Traps:

A (One): There are two visible contact points with the $x$ -axis, not just one.
C (Three): The double root at $x \approx 1$ counts as one $x$ -value, not two. The question asks for how many values of $x$ , not how many roots counting multiplicity.
D (Four): A degree-3 polynomial has at most 3 real roots, so 4 is impossible.

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