Given Roots and Possible Factors

Oct 13, 2025 by windailys.com 122 views

Understanding Polynomial Functions and Roots

A polynomial function is an expression consisting of variables and coefficients combined using only addition, subtraction, and multiplication, and non-negative integer exponents. The roots of a polynomial function are the values of x that make the function equal to zero. In other words, if a polynomial function f(x) has a root r, then f(r) = 0.

Complex Roots and Factors

When a polynomial function has a complex root, it means that the root is a complex number, which is a number that can be expressed in the form a + bi, where a and b are real numbers and i is the imaginary unit, which satisfies the equation i^2 = -1. Complex roots always come in conjugate pairs, which means that if a polynomial function has a complex root a + bi, then it also has a complex root a - bi.

Given Roots and Possible Factors

We are given that the polynomial function f(x) has roots 4 - 13i and 5. Since 4 - 13i is a complex root, its conjugate 4 + 13i must also be a root of the function. Therefore, the factors of f(x) must be of the form (x - (4 - 13i)) and (x - (4 + 13i)).

Analyzing the Options

Let's analyze the options given:

A. (x + (13 - 4i)) - This option is incorrect because the root 4 - 13i is a complex number, and its conjugate 4 + 13i is the correct root to pair with it.

B. (x - (13 + 4i)) - This option is incorrect because the root 4 - 13i is a complex number, and its conjugate 4 + 13i is the correct root to pair with it.

C. (x + (4 + 13i)) - This option is incorrect because the root 4 - 13i is a complex number, and its conjugate 4 + 13i is the correct root to pair with it.

D. (x - (4 + 13i)) - This option is correct because it correctly pairs the complex root 4 - 13i with its conjugate 4 + 13i.

Conclusion

In conclusion, if a polynomial function f(x) has roots 4 - 13i and 5, then a factor of f(x) must be of the form (x - (4 + 13i)). This is because complex roots always come in conjugate pairs, and the correct pair of roots is 4 - 13i and 4 + 13i.

Final Answer

The final answer is D. (x - (4 + 13i)).

Understanding Polynomial Functions and Roots

Complex Roots and Factors

Q&A Session

Q: What is the difference between a root and a factor of a polynomial function?

A: A root of a polynomial function is a value of x that makes the function equal to zero. A factor of a polynomial function is an expression that, when multiplied by another expression, gives the original polynomial function.

Q: Why do complex roots always come in conjugate pairs?

A: Complex roots always come in conjugate pairs because the coefficients of the polynomial function are real numbers. When a polynomial function has a complex root a + bi, its conjugate a - bi must also be a root of the function, because the coefficients of the polynomial function are real numbers.

Q: How do I find the factors of a polynomial function if I know its roots?

A: If you know the roots of a polynomial function, you can find its factors by using the fact that the factors of a polynomial function are of the form (x - r), where r is a root of the function.

Q: What is the relationship between the roots of a polynomial function and its factors?

A: The roots of a polynomial function are the values of x that make the function equal to zero. The factors of a polynomial function are expressions that, when multiplied together, give the original polynomial function. The roots of a polynomial function are the values of x that make each factor equal to zero.

Q: Can a polynomial function have a complex root and a real root at the same time?

A: Yes, a polynomial function can have a complex root and a real root at the same time. For example, the polynomial function f(x) = (x - 2)(x - 3i) has a real root x = 2 and a complex root x = 3i.

Q: How do I determine if a polynomial function has a complex root or a real root?

A: You can determine if a polynomial function has a complex root or a real root by looking at the coefficients of the polynomial function. If the coefficients are real numbers, then the polynomial function can have complex roots. If the coefficients are complex numbers, then the polynomial function can have real roots.

Conclusion

In conclusion, if a polynomial function has roots 4 - 13i and 5, then a factor of f(x) must be of the form (x - (4 + 13i)). This is because complex roots always come in conjugate pairs, and the correct pair of roots is 4 - 13i and 4 + 13i.

Final Answer

The final answer is D. (x - (4 + 13i)).