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Chapter 2: Problem 98

Use synthetic division to verify the upper and lower bounds of the real zeros of \(f\) \(f(x)=2 x^{4}-8 x+3\) (a) Upper: \(x=3\) (b) Lower: \(x=-4\)

Short Answer

Expert verified
To ascertain whether \(x = 3\) and \(x = -4\) are upper and lower bounds respectively of the real zeros of \(f(x) = 2x^4 - 8x + 3\), synthetic division needs to be applied twice, and the resulting sign patterns examined. If the pattern adheres to the Upper and Lower Bound Rules, then the bounds are verified. Without performing the actual synthetic division, we can't say definitively whether these are accurate bounds.

Step by step solution

01

Synthetic Division with Upper Bound

Perform synthetic division on the function \(f(x) = 2x^4 - 8x + 3\) using the potential upper bound \(x = 3\). Set up the synthetic division by writing down coefficients of \(f(x)\), which are \[2, 0, 0, -8, 3\] (zero coefficients for missing terms \(x^3\) and \(x^2\) have to be included), and then perform synthetic division.
02

Apply the Upper Bound Rule

After performing synthetic division, interpret the resulting sign pattern. If the signs of the resulting numbers alternate or are all positive, then \(x = 3\) is indeed an upper bound of the real zeros of \(f(x)\). If not, it is not an upper bound.
03

Synthetic Division with Lower Bound

Repeat synthetic division on the function \(f(x) = 2x^4 - 8x + 3\) now using the potential lower bound \(x = -4\). Again, set up the synthetic division and execute it.
04

Apply the Lower Bound Rule

Lastly, interpret the resulting sign pattern. If the signs of the resulting numbers alternate beginning with a negative or are all negative, then \(x = -4\) is indeed a lower bound of the real zeros of \(f(x)\). If not, it is not a lower bound.

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Most popular questions from this chapter

Write the polynomial as the product of linear factors and list all the zeros of the function. $$g(x)=2 x^{3}-x^{2}+8 x+21$$

Determine (if possible) the zeros of the function \(g\) when the function \(f\) has zeros at \(x=r_{1}, x=r_{2},\) and \(x=r_{3}\) $$g(x)=f(-x)$$

Find all real zeros of the function. $$f(x)=4 x^{3}-3 x-1$$

Write the polynomial (a) as the product of factors that are irreducible over the rationals, (b) as the product of linear and quadratic factors that are irreducible over the reals, and (c) in completely factored form. \(f(x)=x^{4}-3 x^{3}-x^{2}-12 x-20\) (Hint: One factor is \(\left.x^{2}+4 .\right)\)

Use the information in the table to answer each question. $$\begin{array}{|c|c|} \hline \text { Interval } & \text { Value of } f(x) \\\\\hline(-\infty,-2) & \text { Positive } \\\\\hline(-2,1) & \text { Negative } \\\\\hline(1,4) & \text { Negative } \\\\\hline(4, \infty) & \text { Positive } \\\\\hline\end{array}$$ (a) What are the three real zeros of the polynomial function \(f ?\) (b) What can be said about the behavior of the graph of \(f\) at \(x=1 ?\) (c) What is the least possible degree of \(f ?\) Explain. Can the degree of \(f\) ever be odd? Explain. (d) Is the leading coefficient of \(f\) positive or negative? Explain. (e) Sketch a graph of a function that exhibits the behavior described in the table.
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