Answer:
- degree 4
- odd multiplicity: -1, 4
- even multiplicity: -4
- f(x) = -(x +4)²(x +1)(x -4)
Explanation:
You want the zeros, their multiplicity, the degree, and the equation of the polynomial shown in the graph.
Axis crossing
Consider a zero at x = 1, for example. This corresponds to a factor of (x-1).
For values of x > 1, this factor will be positive. For values of x < 1, this factor will be negative. The sign of the factor changes at x=1. The same will be true for any odd power of this factor. (The graph crosses the x-axis.)
Consider the same zero with even multiplicity. The factors corresponding to that zero will have even degree: (x-1)² or (x-1)⁴, for example. For values of x > 1, the sign of this term will be positive. For values of x < 1, the sign of this term will also be positive. The sign of the term does not change at x=1 for even multiplicity. (The graph does not cross the x-axis.)
Degree
The degree of the polynomial will be the sum of the multiplicities of its zeros. From left to right, the zeros and their multiplicities are ...
- -4, multiplicity 2
- -1, multiplicity 1
- +4, multiplicity 1
The sum of the multiplicities is 2+1+1 = 4, so the least possible degree is 4.
Odd multiplicity
From the above list, the zeros with odd multiplicity are ...
-1, 4
Even multiplicity
The zero with even multiplicity is ...
-4
Equation
A polynomial of even degree has a U-shape, opening upward or downward. This graph opens downward, indicating the leading coefficient is negative. The least product of the found factors will be ...
f(x) = -(x +4)²(x +1)(x -4)
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Additional comment
The attached graph shows the function with a scale factor that makes the graph approximate the one shown. With a leading coefficient of -1, the local minimum is considerably less than the value shown in the problem statement. Your function is constrained to have a leading coefficient of ±1, so the answer here has it as -1.