Question
Question 1 (2 points) Using the Fundamental Theorem of Algebra, how many roots/solutions should the polynomial have? \( f(x)=x^{4}+2 x^{3}-6 x^{2}+4 x-16 \) 4 6 3
Ask by Carrillo Wood. in the United States
Feb 03,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The polynomial has 4 roots/solutions.
Solution
Using the **Fundamental Theorem of Algebra**, a polynomial of degree \( n \) has exactly \( n \) roots (including complex and multiple roots).
For the polynomial \( f(x) = x^{4} + 2x^{3} - 6x^{2} + 4x - 16 \):
- The degree of the polynomial is **4**.
Therefore, it has **4 roots/solutions**.
**Answer:** 4
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution Beyond the Answer
The Fundamental Theorem of Algebra tells us that a polynomial of degree n will have exactly n roots, counting multiplicities. In this case, your polynomial \( f(x) = x^{4} + 2x^{3} - 6x^{2} + 4x - 16 \) is a degree 4 polynomial, which means it will have 4 roots or solutions, either real or complex. Now, here’s a fun fact: the number of roots can be visualized with a graph! When you plot the polynomial, the intersections with the x-axis represent the real roots. So, not only does the theorem give you a number, but it's also a gateway to exploring the polynomial's behavior visually.
