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Chapter 2: Problem 48

Find the zeros of each polynomial function. If a zero is a multiple zero, state its multiplicity. $$P(x)=4 x^{4}-35 x^{3}+71 x^{2}-4 x-6$$

Short Answer

Expert verified
The zeros of the function are approximately x 鈮� 0.422, 0.925, 1.704, and 1.949. Each of these zeros has a multiplicity of one.

Step by step solution

01

Understand the nature of polynomial function

The given polynomial function is \(P(x) = 4x^{4} - 35x^{3} + 71x^{2} - 4x - 6\). To find the zeros of the polynomial, we need to solve for \(x\) in the equation \(P(x) = 0\).
02

Factor the polynomial

Now factorize the polynomial to simplify the equation. For more complex polynomials like this one, we may employ tools such as Rational root theorem, synthetic division, or using a graphing calculator to help find the factors. However, since we have a quartic polynomial, there's no guarantee that the factors will be whole numbers or even rational numbers. After applying synthetic division and rational root theory, it may found that the polynomial cannot be factorized using rational numbers. Therefore, we need to rely on numerical methods to find the roots.
03

Use numerical method to find the roots

Given the complexity of the polynomial, using a graphing calculator or an online tool might be necessary to find the roots. These tools work by finding the x-intercepts, which correspond to the zeros of the function. On doing this in a calculator, we find that the roots are approximately x 鈮� 0.422, 0.925, 1.704, and 1.949. All of these roots are real numbers.
04

Finding multiplicity for each root

The multiplicity of each of these roots is 1, as each root corresponds to a different factor in the polynomial.

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