Understanding the Given Roots

Oct 13, 2025 by windailys.com 6 views

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Introduction

In mathematics, polynomial functions are a fundamental concept in algebra, and understanding their roots is crucial for solving various problems. Given a polynomial function, we can determine its roots, which are the values of x that satisfy the equation f(x) = 0. In this article, we will analyze the roots of the polynomial function f(x) = x^5 - 8x^4 + 21x^3 - 12x^2 - 22x + 20, where three roots are already known: -1, 1, and 3+i.

Understanding the Given Roots

The given roots are -1, 1, and 3+i. Since 3+i is a complex root, we know that the polynomial function has at least one complex root. The presence of complex roots indicates that the polynomial function has a non-real root.

Determining the Number and Nature of Roots

To determine the number and nature of all the roots of the function, we need to consider the following:

Complex Conjugate Roots: If a polynomial function has a complex root, it must also have the complex conjugate of that root as a root. In this case, the complex conjugate of 3+i is 3-i.
Real Roots: The remaining roots must be real. Since we have three real roots already (-1, 1, and 3+i is not a real root), we need to find the remaining real roots.
Total Number of Roots: The total number of roots of a polynomial function is equal to its degree. In this case, the degree of the polynomial function is 5.

Applying the Rational Root Theorem

The rational root theorem states that if a rational number p/q is a root of the polynomial function f(x), then p must be a factor of the constant term, and q must be a factor of the leading coefficient. In this case, the constant term is 20, and the leading coefficient is 1.

Finding the Remaining Real Roots

Using the rational root theorem, we can find the remaining real roots by testing the factors of 20 as potential roots. The factors of 20 are ±1, ±2, ±4, ±5, ±10, and ±20.

Testing Potential Roots

We can test these potential roots by substituting them into the polynomial function and checking if the result is equal to zero.

Conclusion

In conclusion, the polynomial function f(x) = x^5 - 8x^4 + 21x^3 - 12x^2 - 22x + 20 has a total of 5 roots. The three given roots are -1, 1, and 3+i. The remaining two roots are real and can be found using the rational root theorem. The complex conjugate of 3+i is 3-i, which is also a root of the polynomial function.

Final Answer

The final answer is that the polynomial function f(x) = x^5 - 8x^4 + 21x^3 - 12x^2 - 22x + 20 has 5 roots: -1, 1, 3+i, 3-i, and another real root.

Roots of the Polynomial Function

The roots of the polynomial function f(x) = x^5 - 8x^4 + 21x^3 - 12x^2 - 22x + 20 are:

-1
1
3+i
3-i
Another real root (to be determined)

Determining the Remaining Real Root

To determine the remaining real root, we can use the rational root theorem and test the factors of 20 as potential roots.

Testing the Factors of 20

The factors of 20 are ±1, ±2, ±4, ±5, ±10, and ±20. We can test these potential roots by substituting them into the polynomial function and checking if the result is equal to zero.

Finding the Remaining Real Root

After testing the potential roots, we find that the remaining real root is 2.

Conclusion

In conclusion, the polynomial function f(x) = x^5 - 8x^4 + 21x^3 - 12x^2 - 22x + 20 has a total of 5 roots: -1, 1, 3+i, 3-i, and 2.

Final Answer

The final answer is that the polynomial function f(x) = x^5 - 8x^4 + 21x^3 - 12x^2 - 22x + 20 has 5 roots: -1, 1, 3+i, 3-i, and 2.

Roots of the Polynomial Function

The roots of the polynomial function f(x) = x^5 - 8x^4 + 21x^3 - 12x^2 - 22x + 20 are:

-1
1
3+i
3-i
2

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Q: What is the degree of the polynomial function f(x) = x^5 - 8x^4 + 21x^3 - 12x^2 - 22x + 20?

A: The degree of the polynomial function f(x) = x^5 - 8x^4 + 21x^3 - 12x^2 - 22x + 20 is 5, which means it has 5 roots.

Q: What are the given roots of the polynomial function f(x) = x^5 - 8x^4 + 21x^3 - 12x^2 - 22x + 20?

A: The given roots of the polynomial function f(x) = x^5 - 8x^4 + 21x^3 - 12x^2 - 22x + 20 are -1, 1, and 3+i.

Q: Is the polynomial function f(x) = x^5 - 8x^4 + 21x^3 - 12x^2 - 22x + 20 a real or complex polynomial function?

A: The polynomial function f(x) = x^5 - 8x^4 + 21x^3 - 12x^2 - 22x + 20 is a complex polynomial function because it has a complex root (3+i).

Q: What is the complex conjugate of the root 3+i?

A: The complex conjugate of the root 3+i is 3-i.

Q: How many real roots does the polynomial function f(x) = x^5 - 8x^4 + 21x^3 - 12x^2 - 22x + 20 have?

A: The polynomial function f(x) = x^5 - 8x^4 + 21x^3 - 12x^2 - 22x + 20 has 2 real roots: -1 and 1.

Q: What is the remaining real root of the polynomial function f(x) = x^5 - 8x^4 + 21x^3 - 12x^2 - 22x + 20?

A: The remaining real root of the polynomial function f(x) = x^5 - 8x^4 + 21x^3 - 12x^2 - 22x + 20 is 2.

Q: What is the total number of roots of the polynomial function f(x) = x^5 - 8x^4 + 21x^3 - 12x^2 - 22x + 20?

A: The total number of roots of the polynomial function f(x) = x^5 - 8x^4 + 21x^3 - 12x^2 - 22x + 20 is 5.

Q: What are the roots of the polynomial function f(x) = x^5 - 8x^4 + 21x^3 - 12x^2 - 22x + 20?

A: The roots of the polynomial function f(x) = x^5 - 8x^4 + 21x^3 - 12x^2 - 22x + 20 are -1, 1, 3+i, 3-i, and 2.

Q: How can we determine the remaining real root of the polynomial function f(x) = x^5 - 8x^4 + 21x^3 - 12x^2 - 22x + 20?

A: We can determine the remaining real root of the polynomial function f(x) = x^5 - 8x^4 + 21x^3 - 12x^2 - 22x + 20 by using the rational root theorem and testing the factors of 20 as potential roots.

Q: What is the rational root theorem?

A: The rational root theorem states that if a rational number p/q is a root of the polynomial function f(x), then p must be a factor of the constant term, and q must be a factor of the leading coefficient.

Q: How can we apply the rational root theorem to find the remaining real root of the polynomial function f(x) = x^5 - 8x^4 + 21x^3 - 12x^2 - 22x + 20?

A: We can apply the rational root theorem by testing the factors of 20 as potential roots and substituting them into the polynomial function to check if the result is equal to zero.

Q: What are the factors of 20?

A: The factors of 20 are ±1, ±2, ±4, ±5, ±10, and ±20.

Q: How can we test the factors of 20 as potential roots?

A: We can test the factors of 20 as potential roots by substituting them into the polynomial function and checking if the result is equal to zero.

Q: What is the remaining real root of the polynomial function f(x) = x^5 - 8x^4 + 21x^3 - 12x^2 - 22x + 20?

A: The remaining real root of the polynomial function f(x) = x^5 - 8x^4 + 21x^3 - 12x^2 - 22x + 20 is 2.