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Chapter 0: Problem 98

Writing Use a graphing utility to graph the polynomial functions \(p_{1}(x)=x^{3}-x+1\) and \(p_{2}(x)=x^{3}-x .\) How many zeros does each function have? Is there a cubic polynomial that has no zeros? Explain.

Short Answer

Expert verified
Function \(p_1(x) = x^3 - x + 1\) and \(p_2(x) = x^3 - x\) have real zeros at the points where they cross the x-axis. And yes, according to the Fundamental theorem of Algebra, a cubic polynomial always has three roots or zeros, which can be real or complex.

Step by step solution

01

Graph the polynomials

Use a graphing utility to accurately plot the two functions \(p_1(x) = x^3 - x + 1\) and \(p_2(x) = x^3 - x\).
02

Identify the zeros

On your graph, identify the point(s) where the graph crosses the x-axis. These crossing points are the zeros of the function, which means they are the solution to the equation when the function equals zero.
03

Count the numbers of zeros

Count the numbers of these crossing points for each function.
04

Understand the characteristics of cubic polynomials

A cubic polynomial is a polynomial of degree 3. Fundamental theorem of algebra states that any polynomial of degree n has exactly n roots or zeros (which could be real or complex numbers). Thus, a cubic polynomial has three roots, but this does not necessarily mean they are all real. They could be complex roots.
05

Answer the question

It is not possible for a cubic polynomial to have no zeros; it will always have three zeros, which could be either real or complex numbers, depending on the specific polynomial.

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Most popular questions from this chapter

Deciding Whether an Equation Is a Function In Exercises \(47-50\) , determine whether \(y\) is a function of \(x .\) $$ x^{2}+y=16 $$

Sketching a Graph of a Function In Exercises \(33-40\) , sketch a graph of the function and find its domain and range. Use a graphing utility to verify your graph. $$ h(x)=\sqrt{x-6} $$

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