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Chapter 2: Problem 51

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

Short Answer

Expert verified
The roots of the polynomial \(P(x)=x^{3}-3x-2\) are \(x = -1\) and \(x = 2\), each with a multiplicity of 1.

Step by step solution

01

Set the Polynomial Equal to Zero

Solve the following equation \(P(x) =0 \) which gives \(x^3 -3x -2 =0\).
02

Applying Rational Root Theorem

According to the Rational Root Theorem, the possible rational roots of the polynomial are factors of the constant term divided by factors of the leading coefficient. Here since the leading coefficient is 1, we need to consider factors of -2 which are 卤1, 卤2. When testing these values we find \(x = -1\) and \(x = 2\) satisfy the equation.
03

Verify The Roots and Check Multiplicity

Substitute \(x = -1\) and \(x = 2\) into the original equation to verify the truth of the identities. Since both verify, we have two roots. They are not repeated, so their multiplicity is 1.

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