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Chapter 2: Problem 57

Find the zeros of each polynomial function. If a zero is a multiple zero, state its multiplicity. $$P(x)=2 x^{4}-17 x^{3}+4 x^{2}+35 x-24$$

Short Answer

Expert verified
The zeros of the polynomial \(P(x)=2x^{4}-17x^{3}+4x^{2}+35x-24\) are \(x = 1, x = 3/2, x = 4, and x = -2\), each with a multiplicity of 1.

Step by step solution

01

Apply The Zero Product Property

According to the Zero Product Property, if a product of factors is zero, then at least one of the factors must be zero. So, start by factoring \(P(x) = 2x^{4}-17x^{3}+4x^{2}+35x-24\) to set each of the factors equal to zero.
02

Factor The Polynomial

The next step is to factor the polynomial \(P(x)\). By using synthetic division or another factoring method, it can be found that the polynomial factors as \(P(x) = (x-1)(2x-3)(x-4)(x+2)\).
03

Set Each Factor Equal To Zero

According to the Zero Product Property explained in Step 1, if any of the factors is zero then the entire equation will be zero. Therefore, set each factor equal to zero and solve for x: \(x-1 = 0, 2x-3 = 0, x-4 = 0, x+2 = 0\). This gives the solutions \(x = 1, x = 3/2, x = 4, x = -2\).
04

Determine The Multiplicity Of Each Zero

The multiplicity of a root is the exponent on its corresponding factor in the factored form of the polynomial. In this case, all of the exponents on the factors of the polynomial are 1, so each of the roots has a multiplicity of 1.

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