Finding the Remaining Zeros

Oct 13, 2025 by miodeya.com 68 views

Introduction

In this article, we will discuss how to find the remaining zeros of a polynomial given some of its zeros. The polynomial in question is of degree 5, and we are given that its coefficients are real numbers. We are also given three of its zeros: -6, i, and 4i. Our goal is to find the remaining two zeros of the polynomial.

Complex Conjugate Zeros

The given zeros are -6, i, and 4i. Notice that i and 4i are complex conjugates of each other. In general, if a polynomial has real coefficients and a complex zero, then its complex conjugate is also a zero of the polynomial. This is because complex conjugates have the same real part and opposite imaginary parts, which means that they are symmetric with respect to the real axis.

Quadratic Factors

Since we know that i and 4i are zeros of the polynomial, we can write a quadratic factor that has these zeros. The quadratic factor is given by:

x^2 + 16

This quadratic factor has zeros at i and 4i, and it is a factor of the polynomial f(x).

Linear Factors

We are also given that -6 is a zero of the polynomial. This means that we can write a linear factor that has this zero. The linear factor is given by:

(x + 6)

This linear factor has a zero at -6, and it is a factor of the polynomial f(x).

Factoring the Polynomial

We can now factor the polynomial f(x) using the quadratic and linear factors that we have found. We can write:

f(x) = (x + 6)(x^2 + 16)

This is a factored form of the polynomial f(x).

Finding the Remaining Zeros

To find the remaining zeros of the polynomial, we need to find the zeros of the quadratic factor x^2 + 16. We can do this by setting the quadratic factor equal to zero and solving for x:

x^2 + 16 = 0

x^2 = -16

x = \pm \sqrt{-16}

x = \pm 4i

We have already found that i and 4i are zeros of the polynomial, so we can ignore these solutions. However, we can see that the quadratic factor x^2 + 16 has two zeros: -4i and 4i. These are the remaining zeros of the polynomial.

Conclusion

In this article, we have discussed how to find the remaining zeros of a polynomial given some of its zeros. We have used the fact that complex conjugates are zeros of the polynomial to find a quadratic factor, and we have used the fact that linear factors have a single zero to find a linear factor. We have then factored the polynomial using these factors and found the remaining zeros of the polynomial.

The Remaining Zeros

The remaining zeros of the polynomial f(x) are -4i and 4i.

Final Answer

$The final answer is: $\boxed{-4i, 4i}$<br/> **Q&A: Finding the Remaining Zeros of a Polynomial**$

Introduction

In our previous article, we discussed how to find the remaining zeros of a polynomial given some of its zeros. We used the fact that complex conjugates are zeros of the polynomial to find a quadratic factor, and we used the fact that linear factors have a single zero to find a linear factor. We then factored the polynomial using these factors and found the remaining zeros of the polynomial.

Q&A

Q: What is the significance of complex conjugates in finding the remaining zeros of a polynomial?

A: Complex conjugates are significant in finding the remaining zeros of a polynomial because they are zeros of the polynomial. If a polynomial has real coefficients and a complex zero, then its complex conjugate is also a zero of the polynomial.

Q: How do I find the quadratic factor of a polynomial given some of its zeros?

A: To find the quadratic factor of a polynomial given some of its zeros, you need to use the fact that complex conjugates are zeros of the polynomial. If you know that a complex number and its conjugate are zeros of the polynomial, then you can write a quadratic factor that has these zeros.

Q: What is the difference between a quadratic factor and a linear factor?

A: A quadratic factor is a quadratic expression that has zeros at the given complex numbers, while a linear factor is a linear expression that has a single zero at the given complex number.

Q: How do I factor a polynomial using the quadratic and linear factors that I have found?

A: To factor a polynomial using the quadratic and linear factors that you have found, you need to multiply the factors together. For example, if you have found a quadratic factor x^2 + 16 and a linear factor (x + 6), then you can factor the polynomial f(x) as follows:

f(x) = (x + 6)(x^2 + 16)

Q: What are the remaining zeros of a polynomial?

A: The remaining zeros of a polynomial are the zeros of the polynomial that are not given in the problem. In other words, they are the zeros that you need to find in order to factor the polynomial completely.

Q: How do I find the remaining zeros of a polynomial?

A: To find the remaining zeros of a polynomial, you need to find the zeros of the quadratic factor that you have found. You can do this by setting the quadratic factor equal to zero and solving for x.

Q: What are some common mistakes to avoid when finding the remaining zeros of a polynomial?

A: Some common mistakes to avoid when finding the remaining zeros of a polynomial include:

Not using the fact that complex conjugates are zeros of the polynomial
Not finding the quadratic factor correctly
Not factoring the polynomial correctly
Not checking the solutions to the quadratic equation

Conclusion

In this article, we have discussed some common questions and answers related to finding the remaining zeros of a polynomial. We have used the fact that complex conjugates are zeros of the polynomial to find a quadratic factor, and we have used the fact that linear factors have a single zero to find a linear factor. We have then factored the polynomial using these factors and found the remaining zeros of the polynomial.

Final Answer

The final answer is: $\boxed{-4i, 4i}$