Question
How many roots does the following function have (including complex)? \( f(x)=4 x^{5}-6 x^{2}-3 x+7 \) 5 4 6 6
Ask by Reeves Parry. in the United States
Jan 30,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The function has 5 roots.
Solution
A polynomial of degree \( n \) has exactly \( n \) roots in the complex plane, counting multiplicities. This is known as the **Fundamental Theorem of Algebra**.
Given the function:
\[ f(x) = 4x^5 - 6x^2 - 3x + 7 \]
- The highest power of \( x \) is 5, which means this is a **5th-degree polynomial**.
According to the Fundamental Theorem of Algebra:
- A 5th-degree polynomial will have exactly **5 roots** (including real and complex roots).
**Answer: 5**
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution Mind Expander
The function \( f(x) = 4x^5 - 6x^2 - 3x + 7 \) is a polynomial of degree 5, which means it has exactly 5 roots in the complex number system, counting multiplicities. This is derived from the fundamental theorem of algebra, which states that a polynomial of degree \( n \) has \( n \) roots. Now, since we're dealing with a polynomial, all possible roots (real and complex) can be found, but the number of real roots could be fewer than 5. Analyzing the graph or using numerical methods might help detect the specific nature of these roots!
