Question

In Exercises 25–26, graph each polynomial function. f(x)=2x2(x−1)3(x+2)f(x) = $$2x^2(x$$ - $$1)^3(x + 2)$$

Accepted Answer

Identify the degree and leading coefficient of the polynomial function $$f(x) = 2x^2$(x - 1)^3$(x + 2). To do $$this, first find the degree by adding the exponents of each factor: $$2x^2$$ contributes degree 2, $$(x - 1)^3$ contributes degree 3, and (x + 2$$)$$ contributes degree 1. So, total degree = 2 + 3 + 1$. Determine the leading term by multiplying the leading terms of each factor: from 2x^2$ take 2x^2$, from (x$$ - $$1)^3$$ take $$x^3$$, and from $$(x + 2)$$ take $$x. $$Multiply these to get the leading term, which will help understand the end behavior of the graph. Find the zeros (roots) of the polynomial by setting each factor equal to zero: $$2x^2 = 0$ gives x=0$, (x$$ - $$1)^3 = 0$ gives x=1$, and (x + 2) = 0$ gives x = -2$. Note the multiplicity of each zero: 2 for x=0$, 3 for x=1$, and 1 for x=-2$. Analyze the behavior of the graph at each zero based on its multiplicity: if the multiplicity is even, the graph touches the x-axis and turns around; if odd, it crosses the x-axis. So, at x=0$ (multiplicity 2) the graph touches and turns, at x=1$ (multiplicity 3) it crosses with a flattening effect, and at x=-2$ (multiplicity 1) it crosses the x-axis. Use the leading term and zeros to sketch the graph: start by plotting the zeros on the x-axis, then use the end behavior from the leading term to determine how the graph behaves as x$$ 	o \pm \infty$$. Connect the points smoothly, respecting the behavior at each zero.

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

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Key Concepts

Polynomial Functions and Their Degree

Zeros of a Polynomial and Their Multiplicities

End Behavior of Polynomial Graphs

In Exercises 25–26, graph each polynomial function. f(x)=2x2(x−1)3(x+2)f(x) = 2x^2(x - 1)^3(x + 2)