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Chapter 2: Problem 131

Sketch the graph of a fifth-degree polynomial function whose leading coefficient is positive and that has a zero at \(x=3\) of multiplicity 2 .

Short Answer

Expert verified
A fifth-degree polynomial function with a positive leading coefficient having a zero at \(x=3\) of multiplicity 2 would look like an 'S' shape with a curvature change at \(x=3\), with both ends pointing upwards and the graph touching (not crossing) the axis at \(x=3\).

Step by step solution

01

Define Polynomial Function

The function can be defined as \(f(x) = a(x-x_1)^{n_1}(x-x_2)^{n_2}... (x-x_m)^{n_m}\), where \(a\) is the leading coefficient and \(x_i\) are the zeros of the function with multiplicity \(n_i\). From the problem, it can be seen that \(a>0\) (positive leading coefficient) and at \(x=3\) there is a zero with multiplicity 2. So, an example of a function satisfying these conditions is \(f(x) = (x-3)^2 * x^3\).
02

Identify key points

In the function \(f(x) = (x-3)^2 * x^3\), the root points are \(x = 3\) and \(x = 0\), with \(x = 3\) having multiplicity 2 and 0 having multiplicity 3.
03

Draw the function

Map out key points on an x-y graph. Sketch a curve passing through the root at 0 with the curve crossing the x-axis since its multiplicity is an odd number. Then, sketch the curve touching (but not crossing) the x-axis at \(x = 3\) because its multiplicity 2 is an even number and the graph should touch the x-axis at \(x=3\) but not cross it. Since the leading coefficient is a positive number, both ends of the graph point upwards for a fifth degree polynomial function.

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