/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 114 Sketch the graph of a polynomial... [FREE SOLUTION] | 91影视

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Chapter 2: Problem 114

Sketch the graph of a polynomial function that is of fifth degree, has a zero of multiplicity \(2,\) and has a negative leading coefficient. Sketch the graph of another polynomial function with the same characteristics except that the leading coefficient is positive.

Short Answer

Expert verified
To sketch a fifth degree polynomial with a zero of multiplicity 2 and a negative leading coefficient, start from the top, bounce back at the double root and then end downwards. For a similar polynomial with a positive leading coefficient, start from the bottom, bounce back at the double root and then end upwards.

Step by step solution

01

Understand the polynomial characteristics

A polynomial of fifth degree is a polynomial with a variable (usually denoted as \(x\)) raised to the highest power of 5. When the problem states 'has a zero of multiplicity 2', it means that one of the roots of the polynomial (the value which makes the polynomial equal to zero) is repeated twice. This will cause the graph to touch the x-axis at that point, but not cross it. The leading coefficient is the number in front of the greatest degree of the polynomial. If this number is negative, the graph will open downwards and if it is positive, it will open upwards.
02

Sketching the Polynomial with Negative Leading Coefficient

With this information, to sketch a fifth degree polynomial with a zero of multiplicity 2 and a negative leading coefficient, determine where the graph will be touching the x-axis without crossing (this is at zero with multiplicity 2). Then, knowing that the graph should end in the negative direction due to the negative leading coefficient and knowing that the graph is an odd degree (ends in opposite directions), we start from the top, go up while passing through the zeros, bounce back at the double root and then end downwards.
03

Sketching the Polynomial with Positive Leading Coefficient

Now to sketch a similar fifth degree polynomial but with a positive leading coefficient, it will touch the x-axis at the same zero of multiplicity 2 (because the characteristics except the leading coefficient are the same). However, due to the positive leading coefficient, the graph will end in the positive direction. Therefore, again knowing that the graph is an odd degree (ends in opposite directions), this time start from the bottom, go up while passing through the zeros, bounce back at the double root and then end upwards.

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Most popular questions from this chapter

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.

For each function, identify the degree of the function and whether the degree of the function is even or odd. Identify the leading coefficient and whether the leading coefficient is positive or negative. Use a graphing utility to graph each function. Describe the relationship between the degree of the function and the sign of the leading coefficient of the function and the right-hand and left-hand behavior of the graph of the function. (a) \(f(x)=x^{3}-2 x^{2}-x+1\) (b) \(f(x)=2 x^{5}+2 x^{2}-5 x+1\) (c) \(f(x)=-2 x^{5}-x^{2}+5 x+3\) (d) \(f(x)=-x^{3}+5 x-2\) (e) \(f(x)=2 x^{2}+3 x-4\) (f) \(f(x)=x^{4}-3 x^{2}+2 x-1\) (g) \(f(x)=x^{2}+3 x+2\)

Use synthetic division to verify the upper and lower bounds of the real zeros of \(f\) \(f(x)=x^{4}-4 x^{3}+16 x-16\) (a) Upper: \(x=5\) (b) Lower: \(x=-3\)

Find all the zeros of the function. When there is an extended list of possible rational zeros, use a graphing utility to graph the function in order to disregard any of the possible rational zeros that are obviously not zeros of the function. $$f(x)=9 x^{3}-15 x^{2}+11 x-5$$

Find all the zeros of the function. When there is an extended list of possible rational zeros, use a graphing utility to graph the function in order to disregard any of the possible rational zeros that are obviously not zeros of the function. $$f(x)=2 x^{4}+5 x^{3}+4 x^{2}+5 x+2$$
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