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Chapter 4: Problem 51

The following third- and fourth-degree polynomials have a property that makes them relatively easy to graph. Make a complete graph and describe the property. $$f(x)=x^{3}-6 x^{2}-135 x$$

Short Answer

Expert verified
The third-degree polynomial function, $$f(x) = x^{3} - 6x^{2} - 135x$$, is relatively easy to graph because it can be quickly factored, allowing us to find the x-intercepts (x = 0, x = 9, and x = -15) and y-intercept (0, 0) efficiently. By identifying these critical points, we can sketch the graph of the function, which increases as x approaches infinity and decreases as x approaches negative infinity.

Step by step solution

01

Identify the x-intercepts

To find the x-intercepts of the function, we need to solve the equation $$f(x) = x^{3} - 6x^{2} - 135x = 0$$. Start by factoring out x: $$x(x^{2} - 6x - 135) = 0$$ Now, solve the quadratic equation $$x^{2} - 6x - 135 = 0$$ using the quadratic formula, $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$: $$x = \frac{-(-6) \pm \sqrt{(-6)^2 - 4(1)(-135)}}{2(1)}$$ After solving the equation, we get the x-intercepts as x = 0, x = 9, and x = -15.
02

Find the y-intercept

To find the y-intercept, we need to find the value of f(x) when x = 0: $$f(0) = (0)^3 - 6(0)^2 - 135(0) = 0$$ So, the y-intercept is at the point (0, 0).
03

Identify the local minimum and maximum (if they exist)

Since this is a third-degree polynomial, there won't be both a local minimum and maximum. Therefore, we can skip this step and proceed to graph the polynomial.
04

Graph the polynomial

Now, we can plot the function using the points we found: - x-intercepts at x = 0, x = 9, and x = -15 - y-intercept at (0,0) Using a graphing tool or software, we can obtain the graph for the given function. The polynomial will increase as x approaches infinity and decrease as x approaches negative infinity since it is a third-degree polynomial with a positive leading term.
05

Describe the property that makes the function easy to graph

The property that simplifies the graphing process is that the given polynomial can be easily factored, allowing us to find the x-intercepts and y-intercepts quickly. To sum up, we have graphed the polynomial function $$f(x) = x^{3} - 6x^{2} - 135x$$ and identified the property that simplifies the graphing process. We found x-intercepts at x = 0, x = 9, and x = -15, and a y-intercept at (0,0).

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Most popular questions from this chapter

Evaluate the following limits in terms of the parameters a and b, which are positive real numbers. In each case, graph the function for specific values of the parameters to check your results. $$\lim _{x \rightarrow 0} \frac{a^{x}-b^{x}}{x}$$

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