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Chapter 4: Problem 47

First use the graph to estimate the \(x\) -intercepts. Then use algebra to determine each \(x\) -intercept. If an intercept involves a radical, give that answer as well as a calculator approximation rounded to three decimal places. Be sure to check that your results are consistent with the initial graphical estimates. $$y=x^{3}-3 x^{2}-5 x$$ (GRAPH CAN'T COPY)

Short Answer

Expert verified
The x-intercepts are at \(x = 0\), \(x = -2\), and \(x = 5\).

Step by step solution

01

Estimate the x-intercepts from the graph

Look at the provided graph of the equation \(y = x^3 - 3x^2 - 5x\). Identify the points where the curve crosses the x-axis. These points are the x-intercepts. Make an initial estimate of the x-intercepts based on the visual graph.
02

Set up the algebraic equation

To find the exact x-intercepts, set the equation \(y = x^3 - 3x^2 - 5x\) to zero: \[0 = x^3 - 3x^2 - 5x\] This equation implies solving for the values of \(x\) where the equation equals zero.
03

Factor the equation

Factor the equation \(x^3 - 3x^2 - 5x = 0\) to find its roots. First, factor out the common term \(x\):\[x(x^2 - 3x - 5) = 0\]This gives one solution: \(x = 0\).
04

Use the quadratic formula for the additional factors

Solve the quadratic equation \(x^2 - 3x - 5 = 0\) using the quadratic formula:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]where \(a = 1\), \(b = -3\), and \(c = -5\). Substitute these values in:\[x = \frac{3 \pm \sqrt{(-3)^2 - 4 \, \cdot\, 1 \, \cdot\, (-5)}}{2 \, \cdot\, 1}\]This simplifies to:\[x = \frac{3 \pm \sqrt{49}}{2}\]
05

Solve the quadratic formula

Evaluate the square root and continue solving:\[x = \frac{3 \pm 7}{2}\]This gives two solutions: \(x = \frac{3 + 7}{2} = 5\) and \(x = \frac{3 - 7}{2} = -2\).
06

Verify the solutions

Our solutions for the x-intercepts are \(x = 0\), \(x = -2\), and \(x = 5\). Compare these values to your initial graphical estimate. They should match the points where the graph intersects the x-axis.

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Key Concepts

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

Factoring Polynomials
Factoring polynomials is a crucial algebraic skill that helps us simplify equations and find precise solutions. When you are given a polynomial equation like \(y = x^3 - 3x^2 - 5x\), one effective strategy is to set the equation equal to zero and factor it. Let's break down the process:
  • Identify any common factors in the equation. In our example, \(x\) is common to all terms, so we factor out \(x\): \(x(x^2 - 3x - 5) = 0\).
  • Once you have factored out the common term, you have simplified the equation, revealing potential solutions.
  • The factored equation \(x(x^2 - 3x - 5) = 0\) gives the first solution \(x = 0\).
Factoring helps in breaking down higher degree polynomials into more manageable components, allowing us to solve for each possible real root more easily.
Quadratic Formula
The quadratic formula is an essential tool for solving quadratic equations, especially when they can't be factored easily. In our context, after factoring, we're left with the equation \(x^2 - 3x - 5 = 0\), which is a quadratic equation.Here's how to use the quadratic formula:
  • Understand the formula itself: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).
  • Identify the coefficients: For \(x^2 - 3x - 5 = 0\), \(a = 1\), \(b = -3\), and \(c = -5\).
  • Substitute these values into the formula to solve for \(x\).
When we applied the formula, we found:
  • \(x = \frac{3 \pm \sqrt{49}}{2}\), which simplifies to \(x = 5\) and \(x = -2\).
By using the quadratic formula, we have uncovered the exact x-intercepts beyond simple factorization, especially useful for equations that are not easily factorable.
Algebraic Solutions
Finding algebraic solutions involves manipulating equations to find precise values. You originally set the polynomial equation to zero:\[0 = x^3 - 3x^2 - 5x\]By determining where the expression equals zero through factoring and quadratic solving, you ultimately identify every possible intersection with the x-axis. Consider these steps:
  • After factoring out the common term, simplify the remaining equation to break it down further.
  • Solve any remaining quadratic equations using the quadratic formula if necessary.
  • The goal is to find all solutions analytically, giving you algebraic precision over graphical methods alone.
Algebraic solutions provide the exact values that make the equation true, ensuring accuracy and consistency across various problem-solving strategies.
Graphical Estimation
Graphical estimation is a practical technique used to gain an initial understanding of where the function crosses the x-axis. The graph of \(y = x^3 - 3x^2 - 5x\) is used as a reference to make an educated guess about the x-intercepts.Benefits of graphical estimation include:
  • Visualizing the curve helps identify approximate solutions quickly, even if they are not precise.
  • It can provide a check against algebraic solutions, confirming accuracy.
  • An estimate from the graph gives you a general sense of where the real roots lie.
When you compare graphical estimation with algebraic solutions, they should align, giving you confidence in your findings. Graphical methods, though initial approximations, are invaluable for confirming the robustness of your algebraic work.

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Most popular questions from this chapter

(a) Determine the \(x\) - and \(y\) -intercepts and the excluded regions for the graph of the given function. Specify your results using a sketch similar to Figure \(16(a) .\) In Exercises \(31-34\) you will first need to factor the polynomial. (b) Graph each function. $$y=x^{3}-4 x^{2}-5 x$$

The functions \(f, g,\) and h are defined as follows: $$ f(x)=2 x-3 \quad g(x)=x^{2}+4 x+1 \quad h(x)=1-2 x^{2} $$ In each exercise, classify the function as linear, quadratic, or neither. $$f \circ f$$

First, use a graphing utility to estimate to the nearest one-tenth the maximum value of the function. Then use algebra to determine the exact value, and check that your answer is consistent with the graphical estimate. (a) \(f(x)=\sqrt{-x^{2}+4 x+12}\) (b) \(g(x)=\sqrt[3]{-x^{2}+4 x+12}\) (c) \(h(x)=-x^{4}+4 x^{2}+12\)

(a) An open-top box is to be constructed from a 6 -by- 8 -in. rectangular sheet of tin by cutting out equal squares at each corner and then folding up the resulting flaps. Let \(x\) denote the length of the side of each cutout square. Show that the volume \(V(x)\) is $$V(x)=x(6-2 x)(8-2 x)$$ (b) What is the domain of the volume function in part (a)? [The answer is not \((-\infty, \infty) .]\) (c) Use a graphing utility to graph the volume function, taking into account your answer in part (b). (d) By zooming in on the turning point, estimate to the nearest one-hundredth the maximum volume.

The population \(y\) (in thousands) of a colony of bacteria after \(t\) hr is given by $$ y=(6 t+12) /(2 t+1) $$ where \(t \geq 0\) (a) Find the initial population and the long-term population. Which is larger? (b) Use a graphing utility to graph the population function. Is the function increasing or decreasing? Check that your response here is consistent with your answers in part (a).
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