According to the Fundamental Theorem of Algebra, which polynomial function has exactly 8 roots?
f (x) = (3 x squared minus 4 x minus 5) (2 x Superscript 6 Baseline minus 5)
f (x) = (3 x Superscript 4 Baseline + 2 x) Superscript 4
f (x) = (4 x squared minus 7) cubed
f (x) = (6 x Superscript 8 Baseline minus 4 x Superscript 5 minus 1) (3 x squared minus 4)

According to the **Fundamental Theorem of Algebra**, a polynomial function of degree \( n \) will have exactly \( n \) roots (counting multiplicities). Therefore, we need to analyze the degree of each given polynomial function to determine which has exactly 8 roots.

1. **\( f(x) = (3x^2 - 4x - 5)(2x^6 - 5) \)**

   - The degree of \( 3x^2 - 4x - 5 \) is 2.
   - The degree of \( 2x^6 - 5 \) is 6.
   - Total degree: \( 2 + 6 = 8 \).

   This polynomial has degree 8, so it has **exactly 8 roots**.

2. **\( f(x) = (3x^4 + 2x)^4 \)**

   - The degree of \( 3x^4 + 2x \) is 4.
   - Raising this to the power of 4 gives a total degree of \( 4 \times 4 = 16 \).
   
   This polynomial has degree 16, so it has **16 roots**.

3. **\( f(x) = (4x^2 - 7)^3 \)**

   - The degree of \( 4x^2 - 7 \) is 2.
   - Raising this to the power of 3 gives a total degree of \( 2 \times 3 = 6 \).
   
   This polynomial has degree 6, so it has **6 roots**.

4. **\( f(x) = (6x^8 - 4x^5 - 1)(3x^2 - 4) \)**

   - The degree of \( 6x^8 - 4x^5 - 1 \) is 8.
   - The degree of \( 3x^2 - 4 \) is 2.
   - Total degree: \( 8 + 2 = 10 \).

   This polynomial has degree 10, so it has **10 roots**.

### Conclusion:
The polynomial function that has **exactly 8 roots** is:
\[
f(x) = (3x^2 - 4x - 5)(2x^6 - 5)
\]

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### Related questions:
1. What are the key properties of the Fundamental Theorem of Algebra?
2. How do we determine the degree of a polynomial?
3. What is the significance of multiplicity in polynomial roots?
4. How do complex roots appear in polynomials with real coefficients?
5. Can a polynomial have fewer real roots than its degree?

### Tip:
Always check for multiplicities when counting roots, as a polynomial might have repeated roots that still contribute to the total count.

According to the Fundamental Theorem of Algebra, which polynomial function has exactly 8 roots?
 f(x) = (3x^2 - 4x - 5)(2x^6 - 5)
 f(x) = (3x^4 + 2x)^4
 f(x) = (4x^2 - 7)^3
 f(x) = (6x^8 - 4x^5 - 1)(3x^2 - 4)

Discover which polynomial function has exactly 8 roots according to the Fundamental Theorem of Algebra. Explore the detailed solution by calculating the degree of each function and understanding the connection to polynomial roots. Perfect for students in Grades 9-12 studying algebra and polynomial functions.

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