Problem 29 - Precalculus : Building Concepts and Connections

Question

One zero of each polynomial is given. Use it to express the polynomial as a product of linear factors over the complex numbers. You may have already factored some of these polynomials into linear and irreducible quadratic factors in the previous group of exercises. $$x^{4}+4 x^{3}-x^{2}+16 x-20 ; \text { zero: } x=-5$$

Step-by-Step Solution

Answer

This problem involves an understanding of polynomial division and the relationship between zeros and factors of a polynomial. The solution requires the polynomial be divided by its known factor, to find the remaining polynomial which can be solved to find other zeros. Finally, the original polynomial is expressed as the product of its linear factors.

Step 1: Identify the Factor Associated with the Given Zero

Given \(x = -5\) as a zero of the polynomial, the factor associated with this zero can be written as \(x + 5\).

Step 2: Divide the Given Polynomial by the Known Factor

Use polynomial division or synthetic division to divide the given polynomial \(x^{4}+4 x^{3}-x^{2}+16 x-20\) by \(x + 5\).

Step 3: Solve the Resulting Polynomial for Other Zeros

Solve the cubic polynomial resulting from the division for its zeros. Note: the cubic polynomial can be factored using the rational root theorem, synthetic division or other techniques to find possible roots.

Step 4: Express the Original Polynomial as a Product of Linear Factors

Finally, express the original polynomial as a product of its linear factors. Each factor should have the form \(x - zero\) or \(x + zero\) if the zero is negative.