9514 1404 393
Answer:
- f(-∞) → -∞
- f(∞) → ∞
- odd degree
- 5 real zeros
Step-by-step explanation:
The general shape of "up to the right" tells you this is an odd-degree polynomial with a positive leading coefficient. As such, the value of f(x) will have the same sign as the value of x when x gets large. The end behavior could be described by ...
lim[x → -∞] f(x) → -∞
lim[x → ∞] f(x) → ∞
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The curve crosses the x-axis 3 times and touches it once without crossing. Each crossing is a real zero. Each touch is a a real zero with an even multiplicity. The shape (flatness) of the touch gives a clue as to the multiplicity. (Flatter means higher multiplicity.) Here the shape indicates the zero has a multiplicity of 2.
So, there are 4 distinct real zeros, one with a multiplicity of 2, for a total of 5 real zeros.
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Additional comment
An even-degree polynomial function has a generally U shape. If the leading coefficient is negative, the U is upside down: ∩. An even-degree polynomial will have the limits of f(x) having the same sign, regardless of the sign of x.
