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Chapter 2: Problem 50

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

Short Answer

Expert verified
The zeros of the polynomial function \(P(x)=2x^{4} + 3x^{3} - 4x^{2} - 3x + 2\) are \(x = 1\), \(x = 0.5\), and \(x = -1\) with multiplicities 1, 1, and 2 respectively.

Step by step solution

01

Set the polynomial equals to zero

First, you need to set the polynomial function equal to zero: \(2x^{4} + 3x^{3} - 4x^{2} - 3x + 2 = 0\)
02

Factoring the Polynomial

Now, we need to factor the polynomial in order to solve for \( x \). This might be the trickiest part as it requires some knowledge and practice in polynomial factorization. The factored form of above polynomial is \((x-1)(2x-1)(x+1)^2=0\)
03

Applying the Zero Product Property

Next, apply the zero-product property which states that if the product of multiple factors is zero, then at least one of the factors must be zero. Thus set each factor equal to zero separately, which gives the following equations: \(x-1 = 0, 2x-1 = 0, x+1 = 0\)
04

Solve for x

By solving these equations we can identify the roots of the polynomial: For \(x-1 = 0\), the root is \(x = 1\), for \(2x-1 = 0\), the root is \(x = 0.5\) and for \(x+1 = 0\), the root is \(x = -1\). The root \(x=-1\) has multiplicity 2 because it appears twice in the factorization and other roots (\(1\) and \(0.5\)) have multiplicity 1 as they appear only once in the factorization.

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