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Chapter 2: Problem 67

Write the polynomial as the product of linear factors and list all the zeros of the function. $$f(x)=x^{4}-16$$

Short Answer

Expert verified
The polynomial \(x^{4}-16\) can be expressed as the product of linear factors \((x-2)(x+2)(x^{2}+4)\), and the zeros of the function are \(x=-2\), \(x=2\), \(x=2i\), and \(x=-2i\).

Step by step solution

01

Express the Polynomial in the Product of Linear Factors

First, recognize that the polynomial \(f(x)=x^{4}-16\) is a difference of squares, which can be factored as: \((x^{2})^{2}-(4)^{2}\). This is in the form of \(a^{2} - b^{2}\), which can be factored as \((a-b)(a+b)\). Thus, the polynomial becomes: \((x^{2}-4)(x^{2}+4)\). Doing the factoring one more time for the term \((x^{2}-4)\), we have \((x+2)(x-2)(x^{2}+4)\). Hence, the polynomial is written as the product of linear factors.
02

Find the Zeros of the Polynomial

The zeros of the function can be found by setting the function equal to zero, and solving for x: \(x+2=0\), \(x-2=0\), and \(x^{2}+4=0\). Solving these gives: \(x=-2\), \(x=2\), \(x=2i\), \(x=-2i\). So, the zeros of the function are -2, 2, 2i, and -2i.

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