According To The Fundamental Theorem Of Algebra, How Many Roots Exist For The Polynomial Function?\[$(9x+7)(4x+1)(3x+4)=0\$\]A. 1 Root B. 3 Roots C. 4 Roots D. 9 Roots

The Fundamental Theorem of Algebra is a fundamental concept in algebra that states that every non-constant polynomial equation of degree n has exactly n complex roots. This theorem is a cornerstone in the field of algebra and has far-reaching implications in various areas of mathematics, including calculus, geometry, and number theory.

The Polynomial Function

The given polynomial function is $(9x+7)(4x+1)(3x+4)=0$. To find the number of roots, we need to understand the nature of this function. The function is a product of three linear factors, each of the form $(ax+b)=0$. The roots of the function are the values of x that satisfy each of these linear factors.

Applying the Fundamental Theorem of Algebra

According to the Fundamental Theorem of Algebra, each linear factor $(ax+b)=0$ has exactly one root. Since the given function is a product of three linear factors, we can apply the theorem to each factor separately.

• The first factor $(9x+7)=0$ has exactly one root.
• The second factor $(4x+1)=0$ has exactly one root.
• The third factor $(3x+4)=0$ has exactly one root.

Combining the Roots

Since each linear factor has exactly one root, the total number of roots for the given polynomial function is the sum of the number of roots for each factor. Therefore, the total number of roots is:

1 (from the first factor) + 1 (from the second factor) + 1 (from the third factor) = 3

Conclusion

Based on the Fundamental Theorem of Algebra, the given polynomial function $(9x+7)(4x+1)(3x+4)=0$ has exactly 3 roots. This is because each linear factor has exactly one root, and the total number of roots is the sum of the number of roots for each factor.

Answer

The correct answer is B. 3 roots.

Additional Insights

The Fundamental Theorem of Algebra has far-reaching implications in various areas of mathematics. It provides a powerful tool for solving polynomial equations and has applications in fields such as calculus, geometry, and number theory.

In conclusion, the given polynomial function has exactly 3 roots, as predicted by the Fundamental Theorem of Algebra. This theorem is a fundamental concept in algebra and has numerous applications in various areas of mathematics.

Understanding the Fundamental Theorem of Algebra

The Fundamental Theorem of Algebra is a fundamental concept in algebra that states that every non-constant polynomial equation of degree n has exactly n complex roots. This theorem is a cornerstone in the field of algebra and has far-reaching implications in various areas of mathematics, including calculus, geometry, and number theory.

The Polynomial Function

The given polynomial function is $(9x+7)(4x+1)(3x+4)=0$. To find the number of roots, we need to understand the nature of this function. The function is a product of three linear factors, each of the form $(ax+b)=0$. The roots of the function are the values of x that satisfy each of these linear factors.

Applying the Fundamental Theorem of Algebra

According to the Fundamental Theorem of Algebra, each linear factor $(ax+b)=0$ has exactly one root. Since the given function is a product of three linear factors, we can apply the theorem to each factor separately.

• The first factor $(9x+7)=0$ has exactly one root.
• The second factor $(4x+1)=0$ has exactly one root.
• The third factor $(3x+4)=0$ has exactly one root.

Combining the Roots

Since each linear factor has exactly one root, the total number of roots for the given polynomial function is the sum of the number of roots for each factor. Therefore, the total number of roots is:

1 (from the first factor) + 1 (from the second factor) + 1 (from the third factor) = 3

Conclusion

Based on the Fundamental Theorem of Algebra, the given polynomial function $(9x+7)(4x+1)(3x+4)=0$ has exactly 3 roots. This is because each linear factor has exactly one root, and the total number of roots is the sum of the number of roots for each factor.

Answer

The correct answer is B. 3 roots.

Additional Insights

The Fundamental Theorem of Algebra has far-reaching implications in various areas of mathematics. It provides a powerful tool for solving polynomial equations and has applications in fields such as calculus, geometry, and number theory.

Q: What is the Fundamental Theorem of Algebra?

A: The Fundamental Theorem of Algebra is a fundamental concept in algebra that states that every non-constant polynomial equation of degree n has exactly n complex roots.

Q: What does the theorem say about the number of roots?

A: According to the theorem, every non-constant polynomial equation of degree n has exactly n complex roots.

Q: How does the theorem apply to the given polynomial function?

A: The given polynomial function is $(9x+7)(4x+1)(3x+4)=0$. To find the number of roots, we need to understand the nature of this function. The function is a product of three linear factors, each of the form $(ax+b)=0$. The roots of the function are the values of x that satisfy each of these linear factors.

Q: How many roots does each linear factor have?

A: According to the Fundamental Theorem of Algebra, each linear factor $(ax+b)=0$ has exactly one root.

Q: How many roots does the given polynomial function have?

A: Since each linear factor has exactly one root, the total number of roots for the given polynomial function is the sum of the number of roots for each factor. Therefore, the total number of roots is:

1 (from the first factor) + 1 (from the second factor) + 1 (from the third factor) = 3

Q: What are the implications of the Fundamental Theorem of Algebra?

A: The Fundamental Theorem of Algebra has far-reaching implications in various areas of mathematics. It provides a powerful tool for solving polynomial equations and has applications in fields such as calculus, geometry, and number theory.

Q: Can you provide an example of how the theorem is used in real-world applications?

A: Yes, the Fundamental Theorem of Algebra is used in various real-world applications, such as:

• Calculus: The theorem is used to solve polynomial equations and find the roots of functions.
• Geometry: The theorem is used to find the intersection points of curves and surfaces.
• Number Theory: The theorem is used to study the properties of integers and modular arithmetic.

Q: What are some common misconceptions about the Fundamental Theorem of Algebra?

A: Some common misconceptions about the Fundamental Theorem of Algebra include:

• The theorem only applies to polynomials with real coefficients: The theorem actually applies to polynomials with complex coefficients.
• The theorem only applies to polynomials of degree n: The theorem actually applies to polynomials of any degree.

Q: How can I apply the Fundamental Theorem of Algebra in my own work?

A: To apply the Fundamental Theorem of Algebra in your own work, you can use the following steps:

1. Identify the polynomial equation: Identify the polynomial equation you want to solve.
2. Determine the degree of the polynomial: Determine the degree of the polynomial.
3. Apply the theorem: Apply the Fundamental Theorem of Algebra to find the number of roots.
4. Solve for the roots: Solve for the roots of the polynomial equation.

By following these steps, you can apply the Fundamental Theorem of Algebra in your own work and solve polynomial equations with ease.

Conclusion

The Fundamental Theorem of Algebra is a fundamental concept in algebra that states that every non-constant polynomial equation of degree n has exactly n complex roots. The theorem has far-reaching implications in various areas of mathematics and provides a powerful tool for solving polynomial equations. By understanding the theorem and its applications, you can apply it in your own work and solve polynomial equations with ease.

Additional Resources

For more information on the Fundamental Theorem of Algebra, you can consult the following resources:

• Textbooks: Consult algebra textbooks for a comprehensive understanding of the theorem and its applications.
• Online resources: Consult online resources, such as Khan Academy and MIT OpenCourseWare, for video lectures and practice problems.
• Mathematical software: Use mathematical software, such as Mathematica and Maple, to visualize and solve polynomial equations.

By using these resources, you can deepen your understanding of the Fundamental Theorem of Algebra and its applications in various areas of mathematics.