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Chapter 4: Problem 20

Find the zeros of the function. Then sketch a graph of the function. \(p(x)=x^3-5 x^2-4 x+20\)

Short Answer

Expert verified
The zeros of the function are the solutions of the equation \(x^3-5x^2-4x+20 = 0\). After finding the zeros, plot the function on a graph, marking the zeros as points where the function crosses the x-axis. The function will have the general shape of a cubic function. The detail graph would depend on the actual zeros of the function.

Step by step solution

01

Find the zeros of the function

To find the zeros of the given cubic polynomial \(p(x)=x^3-5x^2-4x+20\), we need to set \(p(x)\) equal to zero and solve for \(x\). So we have: \(x^3-5x^2-4x+20 = 0\). The solutions to this equation are the zeros of the function. This could be done by factoring, the rational roots theorem, or by using synthetic division. In this case, we need to solve the cubic equation, which could be complex without a proper method or tool like a software or calculator that can compute cubic equations.
02

Graph the function

With the zeros in hand, now move onto the graph of the function. The zeros represent the x-values where the function y = \(p(x)\) crosses or touches the x-axis. The function is a cubic function, so it will have the general shape of an 'S' or a 'reverse S'. Mark the points corresponding to the zeros on your graph. Connect these points with a smooth curve. Remember that the curve approaches \(+\infty\) or \(-\infty\) as x approaches \(+\infty\) or \(-\infty\).
03

Interpret the graph

The graph gives a visual representation of the function. Points where the graph intersects the x-axis are indeed the zero of the function. The shape of the graph gives indication of the general behavior of the function.

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Key Concepts

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

Zeros of Polynomial Functions
Zeros of polynomial functions are the values of the variable that make the function equal to zero. These values are solutions to the equation formed when you set the polynomial equal to zero. For the polynomial function given in the exercise, \( p(x) = x^3 - 5x^2 - 4x + 20 \), finding the zeros requires solving the equation \( x^3 - 5x^2 - 4x + 20 = 0 \).

Finding these zeros can involve techniques such as factoring, the Rational Root Theorem, synthetic division, or using graphing technology.
  • Factoring: Break down the polynomial into simpler polynomials that multiply together to give the original polynomial. Once factored, set each factor equal to zero and solve for the variable.
  • Rational Root Theorem: This theorem can suggest possible rational zeros of the polynomial, which can then be tested by substitution.
  • Synthetic Division: This method helps to divide polynomials and find zeros by systematically testing possible solutions.
If the polynomial cannot be easily factored, using graphing calculators or software can provide a visual guide to approximate or find exact zeros.
Graphing Functions
Graphing a polynomial function involves plotting its zeros and analyzing its behavior across the x-y plane. For a cubic function like \( p(x) = x^3 - 5x^2 - 4x + 20 \), the graph will generally form an 'S' shape or a reversed 'S'.

The zeros found earlier are critical in graphing because they indicate where the graph crosses the x-axis. Here are steps for graphing such a function:
  • Identify the zeros: Mark these on the x-axis of your graph.
  • Determine end behavior: Observe how the function behaves as \( x \) approaches \( +\infty \) and \( -\infty \). For cubic functions, one end of the graph will tend to positive infinity while the other end tends to negative infinity, creating an 鈥楽鈥� curve.
  • Plot additional points: Besides the zeros, pick points between and beyond them to understand the function's behavior more clearly.
  • Draw the curve: Use the points plotted to sketch a smooth, continuous curve.
This graph will give a comprehensive view of the function's behavior including its increasing, decreasing intervals, and any local maxima or minima.
Factoring Polynomials
Factoring polynomials is a crucial step when solving polynomial equations and finding their zeros. It involves expressing the polynomial as a product of its factors, which are polynomials of lower degrees.

For a polynomial like \( p(x) = x^3 - 5x^2 - 4x + 20 \), the goal is to write it as \((x - r_1)(x - r_2)(x - r_3)\), where \( r_1, r_2, \) and \( r_3 \) are the zeros of the polynomial. Here are some techniques to help in factoring:
  • Common factor: First, try to find and factor out any common factors for all terms of the polynomial.
  • Group factoring: Group terms to factor them into pairs that can be easily factored.
  • Quadratic Methods: When dealing with a cubic function, sometimes it can be reduced to a quadratic equation using synthetic division, which can then be solved using the quadratic formula.
Effective factoring not only simplifies the solving of polynomial equations but also assists in visualizing the polynomial鈥檚 graph.

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Most popular questions from this chapter

s 27鈥�32, fi nd the product of the binomials \((x-5)(x+2)(x-6)\)

Solve the quadratic equation by factoring. $$ 2 x^2-10 x-72=0 $$

Determine whether each polynomial is factored completely. If not, factor completely. a. \(7 z^4\left(2 z^2-z-6\right)\) b. \((2-n)\left(n^2+6 n\right)(3 n-11)\) c. \(3(4 y-5)\left(9 y^2-6 y-4\right)\)

Use the method of your choice to factor the polynomial completely. Explain your reasoning. $$ 5 x^5-10 x^4-40 x^3 $$

s 27鈥�32, fi nd the product of the binomials \((4-5 x)(1-2 x)(3 x+2)\)
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