Question

In Exercises 25–26, graph each polynomial function. f(x)=−x3(x+4)2(x−1)f(x) = $$-x^3(x$$ + $$4)^2(x-1)$$

Accepted Answer

Identify the degree and leading term of the polynomial function $$f(x) = -x^3$ (x + 4)^2$ (x - 1). To do $$this, expand the powers and multiply the highest degree terms: the degree is the sum of the exponents, and the leading coefficient comes from multiplying the leading coefficients of each factor. Determine the zeros (roots) of the function by setting each factor equal to zero: solve $$x^3 = 0$, (x$$ + $$4)^2 = 0$, and (x - 1) = 0$. These zeros are the x-values where the graph will cross or touch the x-axis. Analyze the multiplicity of each zero: a zero with an odd multiplicity (like 3 or 1) means the graph crosses the x-axis at that zero, while a zero with an even multiplicity (like 2) means the graph touches the x-axis and turns around at that zero. Determine the end behavior of the polynomial by looking at the leading term's degree and sign. Since the degree is odd and the leading coefficient is negative, the graph will fall to the right and rise to the left (or vice versa, depending on the sign). Plot the zeros on the x-axis, use the multiplicity to sketch the behavior at each zero, and use the end behavior to guide the shape of the graph as x$ approaches positive and negative infinity. Connect these points smoothly to complete the graph.

In Exercises 25–26, graph each polynomial function. $f (x) = - x^{3} (x + 4)^{2} (x - 1)$

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Key Concepts

Polynomial Function and Degree

Zeros and Their Multiplicities

End Behavior of Polynomial Functions

In Exercises 25–26, graph each polynomial function. f(x)=−x3(x+4)2(x−1)f(x) = -x^3(x + 4)^2(x-1)