Check the real zeros of a cubic using graphing or substitution

Key takeaway: When a multiple-choice list gives several decimal roots, the fastest route is usually to test which approximations actually make the function near zero.

We start with

$$f(x)=-x^3+4x^2-4.$$

To see which answer choice is plausible, test the suggested zeros. A root should make $f(x)$ close to $0$.

For the values in the prompt:

• $f(-1)=1$
• $f(-1.2)=3.468$
• $f(0.9)\approx -0.031$
• $f(1.2)\approx 0.032$
• $f(3.7)\approx 0.167$

The only values that look like actual roots are near $0.9$, $1.2$, and maybe a value around $3.7$ if the arithmetic is being used loosely. But the polynomial is cubic, so it can have at most 3 real zeros, and the options must match the correct graph behavior.

A better exact check comes from rewriting

$$-x^3+4x^2-4=0 \quad \Longleftrightarrow \quad x^3-4x^2+4=0.$$

Then you can use a graphing calculator or numerical methods to locate the real zeros. The response that best fits the intended zero estimate is the one with 3 real zeros near the approximate values listed in option C.

So the intended answer is C.

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Pitfalls the pros know 👇 If you test only one candidate value and stop there, it is easy to choose the wrong option. Another mistake is trusting a rounded decimal without checking whether it is near a sign change. For polynomials, a tiny sign change around a test point is usually more informative than a single raw evaluation.

What if the problem changes? If the task had asked for exact zeros, you would first factor if possible; otherwise, you’d use numerical methods. If it had asked for the number of real zeros only, sign analysis and the graph shape would be enough without computing decimals.

Tags: numerical roots, sign changes, cubic function