According to the Fundamental Theorem of Algebra, which polynomial function has exactly 6 roots?
f (x) = 5 x Superscript 4 Baseline + 10 x squared + 2
f (x) = 5 x Superscript 5 Baseline + 3 x Superscript 4 Baseline + 12 x cubed + 7 x squared minus 2 x + 15
f (x) = 6 x Superscript 5 Baseline + x cubed minus 4 x squared + x minus 5
f (x) = 7 x Superscript 6 Baseline + 3 x cubed + 12

According to the **Fundamental Theorem of Algebra**, a polynomial of degree \(n\) has exactly \(n\) roots, counting multiplicities. The degree of a polynomial is determined by the highest exponent of \(x\).

Let's analyze each function:

1. \( f(x) = 5x^4 + 10x^2 + 2 \)  
   - Degree: 4  
   - This polynomial has **4 roots**.

2. \( f(x) = 5x^5 + 3x^4 + 12x^3 + 7x^2 - 2x + 15 \)  
   - Degree: 5  
   - This polynomial has **5 roots**.

3. \( f(x) = 6x^5 + x^3 - 4x^2 + x - 5 \)  
   - Degree: 5  
   - This polynomial has **5 roots**.

4. \( f(x) = 7x^6 + 3x^3 + 12 \)  
   - Degree: 6  
   - This polynomial has **6 roots**.

### Answer:  
The polynomial \( f(x) = 7x^6 + 3x^3 + 12 \) has **exactly 6 roots**.

Would you like further details or have any questions?

Here are 5 related questions:
1. What is the Fundamental Theorem of Algebra in detail?
2. How do multiplicities affect the number of roots in a polynomial?
3. Can a polynomial have fewer real roots than its degree suggests?
4. What is the relationship between complex roots and the degree of a polynomial?
5. How can one factor polynomials to find their roots?

**Tip:** Always check the highest degree of a polynomial to determine the number of possible roots.

According to the Fundamental Theorem of Algebra, which polynomial function has exactly 6 roots?
f(x) = 5x^4 + 10x^2 + 2
f(x) = 5x^5 + 3x^4 + 12x^3 + 7x^2 - 2x + 15
f(x) = 6x^5 + x^3 - 4x^2 + x - 5
f(x) = 7x^6 + 3x^3 + 12

Explore how the Fundamental Theorem of Algebra helps determine which polynomial function has exactly 6 roots. Analyze polynomial degrees and their relation to the number of roots. Suitable for students in Grades 9-12, this problem covers essential algebraic concepts and the Fundamental Theorem of Algebra.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation