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Chapter 2: Problem 68

Write the polynomial as the product of linear factors and list all the zeros of the function. $$f(y)=y^{4}-256$$

Short Answer

Expert verified
The polynomial can be written as a product of linear factors as follows: (y-16)(y+16)(y^{2}+256) and the zeros of the function are y=16 and y=-16.

Step by step solution

01

Express the Polynomial as Difference of Squares

Recognize that the given polynomial can be expressed as difference of squares. Arrange it as follows: \(f(y)=(y^{2})^{2}-(16^{2})^{2}\).
02

Apply the Difference of Squares Formula

Apply the difference of squares formula, which states that \(a^{2}-b^{2} = (a+b)(a-b)\). Thus the expression becomes: \(f(y)=(y^{2}-16^{2})(y^{2}+16^{2})\).
03

Apply the Difference of Squares Formula Again

Notice that the first part \(y^{2}-16^{2}\) can be further factored using the difference of squares formula. Thus the expression becomes: \(f(y)=(y-16)(y+16)(y^{2}+256).\)
04

Identify the Zeros of the Function

The zeros in a product of factors occur for the values that make each factor equal to zero. Therefore, setting each factor to zero gives the following roots: y=16, y=-16. Since the remaining factor \(y^{2}+256\) cannot be factored further over the real numbers, it does not contribute any real roots. The zeros of the function are therefore y=16 and y=-16.

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