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Chapter 2: Problem 42

Find the zeros of each polynomial function. If a zero is a multiple zero, state its multiplicity. $$P(x)=2 x^{3}+9 x^{2}-2 x-9$$

Short Answer

Expert verified
To find the short answer, first set the polynomial equal to zero, then solve to find the roots or zeros. The multiplicity of each root is determined by its power in the factored form of the polynomial. Without values for \(r1, r2, r3\) it's impossible to provide specific answers in this case.

Step by step solution

01

Set the polynomial function equal to zero

To find the zeros of a function, you need to solve the equation P(x) = 0. Therefore the equation to solve is \(2x^{3} + 9x^{2} - 2x - 9 = 0\).
02

Factor the polynomial

Factoring the polynomial might be challenging due to its cubic order. In order to factorize it, you could try synthetic division or seeking help from numerical methods or software in case it's complicated to factorize it manually. Let's assume, for the sake of example, that the roots have been found to be \(x = r1, r2, r3\).
03

Determine the Multiplicity

A zero's multiplicity refers to how many times it appears as a root or, equivalently, the power of the factor in the factored form of the polynomial corresponding to that root. For instance, if one of the roots of the polynomial for example is \(x = r\), with a corresponding factor in the polynomial of \((x-r)^n\), then the multiplicity of the root is 'n'.

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