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Chapter 2: Problem 43

Find the zeros of each polynomial function. If a zero is a multiple zero, state its multiplicity. $$P(x)=2 x^{4}-9 x^{3}-2 x^{2}+27 x-12$$

Short Answer

Expert verified
The zeros of the polynomial \(P(x)=2x^{4}-9x^{3}-2x^{2}+27x-12\) are 1, -2, 3/2, -4/2 and each of them has multiplicity 1.

Step by step solution

01

Identify the Polynomial

The given polynomial is \(P(x)=2x^{4}-9x^{3}-2x^{2}+27x-12\)
02

Factorization of the polynomial

Before factoring the polynomial, it is helpful to find potential zeros using the Rational Root Theorem. This theorem suggests that the possible rational roots of the polynomial are the divisors of the constant term (-12 in this case) divided by the divisors of the leading coefficient (2 in this case). Therefore, the possible rational roots are \(\pm1, \pm2, \pm3, \pm4, \pm6, \pm12, \pm 1/2,\pm 3/2\). By substituting these values into the polynomial, it can found that roots are \(x=1, -2, 3/2, -4/2\). We therefore factorize the polynomial as \(P(x)=2(x-1)(x-(-2))(x-3/2)(x-(4/2))\)
03

Find the Zeros and Their Multiplicity

The zeros of the polynomial are the roots we found, 1, -2, 3/2, and -4/2. Each zero appears only once in the factored form, so they are each of multiplicity 1.

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