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Chapter 2: Problem 54

Find the zeros of each polynomial function. If a zero is a multiple zero, state its multiplicity. $$P(x)=x^{3}-2 x+1$$

Short Answer

Expert verified
The zeros of the polynomial function \(P(x)=x^{3}-2 x+1\) are \(x = 0.652704\), \(x = -1.43426 - 0.374165i\) and \(x = -1.43426 + 0.374165i\), all of them with multiplicity 1.

Step by step solution

01

Set the equation to zero

To find the zeros of the function, set the function \(P(x)\) equal to zero. So, solve the equation \(x^{3}-2 x+1=0\).
02

Solve the equation

Solving cubic equations can be complex however there are methods to solve them such as factorization, synthetic division, using cubic formula among others. In this case, factorization or use of cubic formula would be more complex than expected. This equation needs to be solved using numerical methods. If we use any of these we will find (approximately) that \(x = 0.652704\), \(x = -1.43426 -0.374165i\) and \(x = -1.43426 +0.374165i\)
03

State the multiplicity

The multiplicity of a root is the number of times it is repeated. Here, each of the roots occurs only once. So all roots have multiplicity 1.

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Most popular questions from this chapter

The property that the product of conjugates of the form \((a+b i)(a-b i)\) is equal to \(a^{2}+b^{2}\) can be used to factor the sum of two perfect squares over the set of complex numbers. For example, \(x^{2}+y^{2}=(x+y i)(x-y i) .\) In Exercises 71 to \(74,\) factor the binomial over the set of complex numbers. $$4 x^{2}+81$$

Find the zeros of each polynomial function. If a zero is a multiple zero, state its multiplicity. $$P(x)=x^{4}+x^{3}-3 x^{2}-5 x-2$$

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In Exercises 61 to 70 , use the quadratic formula to solve each quadratic equation. $$2 x^{2}+2 x+13=0$$

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