Problem 55 Determine the number of possible... [FREE SOLUTION] (2024)

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Short Answer

Expert verified

Possible positive real zeros: 6, 4, 2, 0. Possible negative real zeros: 1.

Step by step solution

Identify the polynomial

Given function is $ k(x) = -8x^7 + 5x^6 - 3x^4 + 2x^3 - 11x^2 + 4x - 3 $.

Use Descartes' Rule of Signs for positive zeros

According to Descartes' Rule of Signs, the number of positive real zeros of a polynomial is equal to the number of sign changes in the polynomial function or less than that by a multiple of 2. Identify the sign changes in $ k(x) $: - $ -8x^7 $ to $ 5x^6 $ is one sign change.- $ 5x^6 $ to $ -3x^4 $ is another sign change.- $ -3x^4 $ to $ 2x^3 $ is another sign change.- $ 2x^3 $ to $ -11x^2 $ is another sign change.- $ -11x^2 $ to $ 4x $ is another sign change.- $ 4x $ to $ -3 $ is the final sign change.This gives us 6 sign changes.

Determine possible positive zeros

From the 6 sign changes observed in $ k(x) $, the possible numbers of positive real zeros are 6, 4, 2, or 0 (by reducing by multiples of 2).

Use Descartes' Rule of Signs for negative zeros

To determine the number of negative real zeros, we need to evaluate $ k(-x) $: $ k(-x) = -8(-x)^7 + 5(-x)^6 - 3(-x)^4 + 2(-x)^3 - 11(-x)^2 + 4(-x) - 3 $ Simplifying the expression, we get: $ k(-x) = 8x^7 + 5x^6 - 3x^4 - 2x^3 - 11x^2 - 4x - 3 $

Identify the sign changes in k(-x)

Now count the sign changes in $ k(-x) $: - $ 8x^7 $ to $ 5x^6 $ has no sign change.- $ 5x^6 $ to $ -3x^4 $ is a sign change.- $ -3x^4 $ to $ -2x^3 $ has no sign change.- $ -2x^3 $ to $ -11x^2 $ has no sign change.- $ -11x^2 $ to $ -4x $ has no sign change.- $ -4x $ to $ -3 $ has no sign change.Thus, there is 1 sign change.

Determine possible negative zeros

From the 1 sign change observed in $ k(-x) $, the possible number of negative real zeros is 1.

Summarize the possible positive and negative real zeros

The function $ k(x) = -8x^7 + 5x^6 - 3x^4 + 2x^3 - 11x^2 + 4x - 3 $ has possible numbers of positive real zeros as 6, 4, 2, or 0 and possible number of negative real zeros as 1.

Key Concepts

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

Descartes' Rule of Signs

Descartes' Rule of Signs is a useful mathematical tool to determine the possible number of positive and negative real zeros in a polynomial function. It works by examining the sign changes in the polynomial's coefficients when ordered by descending powers of the variable. Each sign change in the polynomial's terms indicates a potential positive real zero. However, the actual number of positive real zeros will either be equal to the number of sign changes or less than that number by an even integer multiple.

Polynomial Function

A polynomial function is an expression involving a sum of powers in one or more variables multiplied by coefficients. An example of a polynomial function is given by the exercise: \[ k(x) = -8x^7 + 5x^6 - 3x^4 + 2x^3 - 11x^2 + 4x - 3 \]These functions are important as they are used in many areas of mathematics and applied science to represent diverse phenomena. For our purpose, understanding the polynomial helps us apply Descartes' Rule of Signs and identify potential roots.

Positive and Negative Roots

Positive and negative roots (or zeros) are the values of the variable that make the polynomial equal to zero. If a polynomial has a positive root, it means that substituting a positive number for the variable yields zero. Similarly, a negative root means a negative number makes the polynomial equal to zero. For the polynomial function $ k(x) $, we can find the possible positive and negative roots using Descartes' Rule of Signs, as shown in the exercise. By counting the sign changes in $ k(x) $, we determine potential positive roots. By evaluating $ k(-x) $ and counting sign changes, we find potential negative roots.

Sign Changes in Polynomials

Sign changes in a polynomial refer to the point where the sign of consecutive coefficients changes as you move across the terms. For example, consider the polynomial function \[ k(x) = -8x^7 + 5x^6 - 3x^4 + 2x^3 - 11x^2 + 4x - 3 \]As you move term by term:

From $ -8x^7 $ to $ 5x^6 $ constitutes a sign change (
From $ 5x^6 $ to $ -3x^4 $, is another.

This continues throughout the polynomial.In this way, we count 6 sign changes for positive roots.
Next, we look at $ k(-x) $: _{\[ k(-x) = 8x^7 + 5x^6 - 3x^4 - 2x^3 - 11x^2 - 4x - 3 \]Count the sign changes in $ k(-x) $:}
From $ 8x^7 $ to $ 5x^6 $ shows no sign change.
From $ 5x^6 $ to $ -3x^4 $, is a sign change.
Thus, there is 1 sign change for negative roots. Understanding sign changes helps us predict possible zero values for the polynomial.

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Most popular questions from this chapter

Graph the function. $$ v(x)=\frac{2 x^{4}}{x^{2}+9} $$An engineer for a food manufacturer designs an aluminum container for a hotdrink mix. The container is to be a right circular cylinder 5.5 in. in height.The surface area represents the amount of aluminum used and is given by$S(r)=2 \pi r^{2}+11 \pi r,$ where $r$ is the radius of the can. a. Graph the function $y=S(r)$ and the line $y=90$ on the viewing window[0,3,1] by [0,150,10] . b. Use the Intersect feature to determine point of intersection of $y=S(r)$and $y=90$. c. Determine the restrictions on $r$ so that the amount of aluminum used is atmost $90 \mathrm{in}^{2}$. Round to 1 decimal place.a. Factor the polynomial over the set of real numbers. b. Factor the polynomial over the set of complex numbers. $$f(x)=x^{4}+8 x^{2}-33$$Write the domain of the function in interval notation. $$ h(a)=\sqrt{a^{2}-5} $$The light from a lightbulb radiates outward in all directions. a. Consider the interior of an imaginary sphere on which the light shines. Thesurface area of the sphere is directly proportional to the square of theradius. If the surface area of a sphere with a $10-\mathrm{m}$ radius is $400\pi \mathrm{m}^{2}$, determine the surface area of a sphere with a$20-\mathrm{m}$ radius. b. Explain how the surface area changed when the radius of the sphereincreased from $10 \mathrm{~m}$ to $20 \mathrm{~m}$. c. Based on your answer from part (b) how would you expect the intensity oflight to change from a point $10 \mathrm{~m}$ from the lightbulb to a point$20 \mathrm{~m}$ from the lightbulb? d. The intensity of light from a light source varies inversely as the squareof the distance from the source. If the intensity of a lightbulb is 200lumen/m $^{2}$ (lux) at a distance of $10 \mathrm{~m}$, determine theintensity at $20 \mathrm{~m}$.

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