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Chapter 4: Problem 39

Sketch by hand the graph of each function. (You may wish to support your answer with a calculator graph.) $$\begin{aligned} P(x) &=2 x^{5}-10 x^{4}+x^{3}-5 x^{2}-x+5 \\ &=(x-5)\left(x^{2}+1\right)\left(2 x^{2}-1\right) \end{aligned}$$

Short Answer

Expert verified
Graph the endpoints and x/y intercepts, crossing the x-axis at x=5, and touching at x=卤鈭�(1/2).

Step by step solution

01

Identify the Degree of the Polynomial

The given polynomial \( P(x) = 2x^5 - 10x^4 + x^3 - 5x^2 - x + 5 \) is a quintic polynomial, which means it is a fifth-degree polynomial. Since its highest power is odd, the graph will have opposing end behaviors: as \( x \to -\infty \), \( P(x) \to -\infty \) and as \( x \to \infty \), \( P(x) \to \infty \).
02

Determine the Roots

The polynomial is factored as \( (x-5)(x^2+1)(2x^2-1) \). The roots of the polynomial are obtained from this factorization: \( x = 5 \) from \( (x-5) \), and \( 2x^2-1 = 0 \), which gives \( x = \pm \sqrt{\frac{1}{2}} \). \( x^2 + 1 \) has no real roots as it only contributes complex roots \( x = \pm i \). The real roots are \( x = 5 \) and \( x = \pm \sqrt{\frac{1}{2}} \).
03

Find the Y-Intercept

The y-intercept occurs where \( x = 0 \). Substitute \( x = 0 \) into the polynomial: \( P(0) = 2(0)^5 - 10(0)^4 + (0)^3 - 5(0)^2 - 0 + 5 = 5 \). Thus, the y-intercept is \( (0, 5) \).
04

Analyze the Behavior at the Roots

At \( x = 5 \), there is a linear factor \( (x-5) \), indicating the graph crosses the x-axis. At \( x = \pm \sqrt{\frac{1}{2}} \), the factor \( (2x^2-1) \) indicates the graph touches and turns around these roots since it comes from a quadratic expression.
05

Sketch the Graph

Using the information from previous steps, plot the x-intercepts at \( x = 5 \) and \( x = \pm \sqrt{\frac{1}{2}} \) on the x-axis and the y-intercept at \( (0, 5) \). Sketch the curve, ensuring to show it crossing the x-axis at \( x = 5 \) and touching but not crossing at \( x = \pm \sqrt{\frac{1}{2}} \). The graph should head downwards from the left, hit the points \( x = \pm \sqrt{\frac{1}{2}} \), and finally go upwards past \( x = 5 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Quintic Polynomial
In simple terms, a quintic polynomial is a polynomial of degree five. This means it has the highest power of five in its expanded form, resulting in a maximum of five roots or zeros. In our given exercise, the quintic polynomial is \( P(x) = 2x^5 - 10x^4 + x^3 - 5x^2 - x + 5 \). When tackling such polynomials, it's helpful to recognize the degree early, as this provides insight into certain properties of the graph:
  • The end behavior is determined by this degree, particularly because it's odd, indicating opposing ends.
  • The degree affects the polynomial's complexity鈥攄egree five means potentially interacting with up to five x-intercepts.
Factoring helps break down the problem into manageable parts, making the polynomial easier to analyze.
Roots of Polynomial
The roots of a polynomial are values of \( x \) for which the polynomial equals zero. In this exercise, the polynomial is factored as \((x-5)(x^2+1)(2x^2-1)\). This factorization is key for identifying the roots:
  • The factor \( (x-5) \) provides a real root at \( x = 5 \).
  • The factor \( 2x^2 - 1 \) gives real roots at \( x = \pm \sqrt{\frac{1}{2}} \).
  • The term \( x^2 + 1 = 0 \) results in complex roots \( x = \pm i \), which are not plotted on a real graph.
Understanding where a polynomial equals zero is crucial for graphing, as these points indicate where the graph crosses or touches the x-axis.
End Behavior of Polynomial
The end behavior of a polynomial describes the direction the graph heads as \( x \) approaches either positive or negative infinity. For odd-degree polynomials like quintics, such behavior mirrors the leading coefficient:
  • If the leading coefficient is positive, as in our case with \( 2x^5 \), the graph will start from the bottom left (as \( x \) approaches \(-\infty\), \( P(x) \to -\infty \)) and finish at the top right (as \( x \) approaches \(+\infty\), \( P(x) \to \infty \)).
  • End behavior is an important part of sketching the graph because it outlines the graph's directional trend beyond the visible roots or intercepts.
When graphing, it's essential to combine this behavior with the roots and y-intercepts for a complete picture.
Graphing Polynomial Functions
Graphing a polynomial function involves several steps that together reveal the function's shape and critical features. Here鈥檚 how to approach sketching our given quintic polynomial:
  • Identify and Plot the Roots: Place the x-intercepts at \( x = 5 \) and \( x = \pm \sqrt{\frac{1}{2}} \) on the x-axis, noting whether they represent crossings or touches based on their multiplicity.
  • Plot the Y-intercept: Find where the polynomial crosses the y-axis. For \( x = 0 \), the y-intercept is at \( (0, 5) \).
  • Consider End Behavior: As discussed, an odd-degree polynomial with a positive leading coefficient will move from the bottom left to top right.
With these pieces, connect them smoothly to reflect the polynomial's continuity and differentiability, ensuring the graph crosses at \( x = 5 \) and merely touches at \( x = \pm \sqrt{\frac{1}{2}} \). This visualization helps understand polynomial behavior more concretely even without precise computational tools.

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Most popular questions from this chapter

RELATING CONCEPTS For individual or group investigation (Exercises \(55-58\) ) The concepts of stretching, shrinking, translating, and reflecting graphs presented earlier can be applied to polynomial functions of the form \(P(x)=x^{n}\). For example. the graph of \(y=-2(x+4)^{4}-6\) can be obtained from the graph of \(y=x^{4}\) by shifting 4 units to the left. stretching vertically by applying a factor of \(2,\) reflecting across the \(x\) -axis, and shifting downwand 6 units, so the graph should resemble the graph to the right. If we expand the expression \(-2(x+4)^{4}-6\) algebraically, we obtain $$ -2 x^{4}-32 x^{3}-192 x^{2}-512 x-518 $$ Thus, the graph of \(y=-2(x+4)^{4}-6\) is the same as that of $$ y=-2 x^{4}-32 x^{3}-192 x^{2}-512 x-518 $$ In Exercises \(55-58,\) two forms of the same polynomial finction are given. Sketch by hand the general shape of the graph of the function and describe the transformations. Then support your answer by graphing it on your calculator in a suitable window. $$\begin{aligned} &y=2(x+3)^{4}-7\\\ &y=2 x^{4}+24 x^{3}+108 x^{2}+216 x+155 \end{aligned}$$

The close relationships among \(x\) -intercepts of a graph of a function, real zeros of the function, and real solutions of the associated equation should, by now. be apparent to you. Consider the graph of the polynomial function \(P\), given by \(Y_{1}=X^{3}-2 X^{2}-11 X+12.\) (GRAPH CANT COPY) If \(P(x)\) is divided by \(x-2,\) what is the remainder? What is \(P(2) ?\)

Use a graphing calculator to find a comprehensive graph and answer each of the following. (a) Determine the domain. (b) Determine all local minimum points, and tell if any is an absolute minimum point. (Approximate coordinates to the nearest hundredth.) (c) Determine all local maximum points, and tell if any is an absolute maximum point. (Approximate coordinates to the nearest hundredth.) (d) Determine the range. (If an approximation is necessary. give it to the nearest hundredth.) (e) Determine all intercepts. For each function, there is at least one \(x\) -intercept that has an integer x-value. For those that are not integers, give approximations to the nearest hundredth. Determine the \(y\) -intercept analytically. (f) Give the open interval(s) over which the function is increasing. (g) Give the open interval(s) over which the function is decreasing. $$P(x)=-3 x^{6}+2 x^{5}+9 x^{4}-8 x^{3}+11 x^{2}+4$$

Use Descartes' rule of signs to determine the possible numbers of positive and negative real zeros for \(P(x) .\) Then use a graph to determine the actual numbers of positive and negative real zeros. $$P(x)=x^{5}+3 x^{4}-x^{3}+2 x+3$$

Use synthetic substitution to determine whether the given number is a zero of the polynomial. $$\sqrt{6} ; \quad P(x)=-2 x^{6}+5 x^{4}-3 x^{2}+270$$
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