Use The Given Zero To Find The Remaining Zeros Of The Function.$ H(x) = 3x^5 + 2x^4 + 129x^3 + 86x^2 - 882x - 588 }$Given Zero { -7i$ $The Remaining Zero(s) Of { H $}$ Is(are) { \square$}$,

Introduction

In mathematics, polynomial functions are a crucial concept in algebra, and understanding their properties is essential for solving various problems. One of the key aspects of polynomial functions is finding their zeros, which are the values of x that make the function equal to zero. Given a zero of a polynomial function, we can use various techniques to find the remaining zeros. In this article, we will explore how to find the remaining zeros of a polynomial function using a given zero.

Understanding Polynomial Functions

A polynomial function is a function that can be written in the form:

${ f(x) = a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0 }$

where $a_n \neq 0$ and $n$ is a non-negative integer. The zeros of a polynomial function are the values of x that make the function equal to zero, i.e., $f(x) = 0$. In other words, the zeros of a polynomial function are the solutions to the equation $f(x) = 0$.

Given Zero: $-7i$

We are given that one of the zeros of the polynomial function $h(x) = 3x^5 + 2x^4 + 129x^3 + 86x^2 - 882x - 588$ is $-7i$. Since the given zero is a complex number, we can use the fact that complex zeros of a polynomial function with real coefficients always come in conjugate pairs.

Complex Conjugate Zeros

The complex conjugate of a complex number $a + bi$ is defined as $a - bi$. In this case, the complex conjugate of $-7i$ is $7i$. Since the given zero is $-7i$, we can conclude that the complex conjugate zero is $7i$.

Finding the Remaining Zeros

Now that we have found one complex conjugate pair of zeros, we can use this information to find the remaining zeros of the polynomial function. To do this, we can use the fact that the product of the zeros of a polynomial function is equal to the constant term of the polynomial function divided by the leading coefficient.

Product of Zeros

The product of the zeros of a polynomial function is equal to the constant term of the polynomial function divided by the leading coefficient. In this case, the constant term of the polynomial function is $-588$, and the leading coefficient is $3$. Therefore, the product of the zeros is:

${ \frac{-588}{3} = -196 }$

Finding the Remaining Zeros

Since we have already found one complex conjugate pair of zeros, we can use the fact that the product of the zeros is equal to $-196$ to find the remaining zeros. Let's assume that the remaining zeros are $r_1$ and $r_2$. Then, we can write:

${ r_1r_2 = -196 }$

Solving for the Remaining Zeros

To solve for the remaining zeros, we can factor the product of the zeros, which is $-196$. We can write:

${ -196 = -14 \times 14 }$

Finding the Remaining Zeros

Since the product of the remaining zeros is $-14 \times 14$, we can conclude that the remaining zeros are $14$ and $-14$.

Conclusion

In this article, we have explored how to find the remaining zeros of a polynomial function using a given zero. We have used the fact that complex zeros of a polynomial function with real coefficients always come in conjugate pairs and the product of the zeros is equal to the constant term of the polynomial function divided by the leading coefficient. We have found that the remaining zeros of the polynomial function $h(x) = 3x^5 + 2x^4 + 129x^3 + 86x^2 - 882x - 588$ are $14$ and $-14$.

Final Answer

The remaining zeros of the polynomial function $h(x) = 3x^5 + 2x^4 + 129x^3 + 86x^2 - 882x - 588$ are $\boxed{14, -14}$.

References

• [1] "Polynomial Functions" by Math Open Reference
• [2] "Complex Conjugate Zeros" by Wolfram MathWorld
• [3] "Product of Zeros" by Purplemath

Glossary

• Polynomial Function: A function that can be written in the form $f(x) = a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0$, where $a_n \neq 0$ and $n$ is a non-negative integer.
• Zero: A value of x that makes the function equal to zero, i.e., $f(x) = 0$.
• Complex Conjugate Zeros: Complex zeros of a polynomial function with real coefficients that always come in conjugate pairs.
• Product of Zeros: The product of the zeros of a polynomial function, which is equal to the constant term of the polynomial function divided by the leading coefficient.
  

Introduction

In our previous article, we explored how to find the remaining zeros of a polynomial function using a given zero. We used the fact that complex zeros of a polynomial function with real coefficients always come in conjugate pairs and the product of the zeros is equal to the constant term of the polynomial function divided by the leading coefficient. In this article, we will answer some frequently asked questions about finding remaining zeros of a polynomial function.

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

A: A zero and a root of a polynomial function are the same thing. They are the values of x that make the function equal to zero, i.e., $f(x) = 0$. The terms "zero" and "root" are often used interchangeably in mathematics.

Q: How do I find the remaining zeros of a polynomial function if I know one zero?

A: To find the remaining zeros of a polynomial function if you know one zero, you can use the fact that complex zeros of a polynomial function with real coefficients always come in conjugate pairs. You can also use the fact that the product of the zeros is equal to the constant term of the polynomial function divided by the leading coefficient.

Q: What is the product of the zeros of a polynomial function?

A: The product of the zeros of a polynomial function is equal to the constant term of the polynomial function divided by the leading coefficient. This is a fundamental property of polynomial functions and is used to find the remaining zeros of a polynomial function.

Q: Can I find the remaining zeros of a polynomial function if I know two zeros?

A: Yes, you can find the remaining zeros of a polynomial function if you know two zeros. You can use the fact that the product of the zeros is equal to the constant term of the polynomial function divided by the leading coefficient to find the remaining zeros.

Q: How do I know if a polynomial function has real or complex zeros?

A: To determine if a polynomial function has real or complex zeros, you can use the fact that complex zeros of a polynomial function with real coefficients always come in conjugate pairs. If the polynomial function has real coefficients and has complex zeros, then the complex zeros must come in conjugate pairs.

Q: Can I find the remaining zeros of a polynomial function if I know a complex zero?

A: Yes, you can find the remaining zeros of a polynomial function if you know a complex zero. You can use the fact that complex zeros of a polynomial function with real coefficients always come in conjugate pairs to find the remaining zeros.

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

A: The zeros of a polynomial function are the values of x that make the function equal to zero, i.e., $f(x) = 0$. The factors of a polynomial function are the expressions that, when multiplied together, give the polynomial function. The zeros of a polynomial function are related to its factors in that the zeros are the values of x that make the factors equal to zero.

Q: Can I find the remaining zeros of a polynomial function if I know a factor of the polynomial function?

A: Yes, you can find the remaining zeros of a polynomial function if you know a factor of the polynomial function. You can use the fact that the zeros of a polynomial function are the values of x that make the factors equal to zero to find the remaining zeros.

Conclusion

In this article, we have answered some frequently asked questions about finding remaining zeros of a polynomial function. We have used the fact that complex zeros of a polynomial function with real coefficients always come in conjugate pairs and the product of the zeros is equal to the constant term of the polynomial function divided by the leading coefficient to find the remaining zeros of a polynomial function. We have also discussed the relationship between the zeros of a polynomial function and its factors.

Final Answer

The remaining zeros of a polynomial function can be found using the fact that complex zeros of a polynomial function with real coefficients always come in conjugate pairs and the product of the zeros is equal to the constant term of the polynomial function divided by the leading coefficient.

References

• [1] "Polynomial Functions" by Math Open Reference
• [2] "Complex Conjugate Zeros" by Wolfram MathWorld
• [3] "Product of Zeros" by Purplemath

Glossary

• Polynomial Function: A function that can be written in the form $f(x) = a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0$, where $a_n \neq 0$ and $n$ is a non-negative integer.
• Zero: A value of x that makes the function equal to zero, i.e., $f(x) = 0$.
• Complex Conjugate Zeros: Complex zeros of a polynomial function with real coefficients that always come in conjugate pairs.
• Product of Zeros: The product of the zeros of a polynomial function, which is equal to the constant term of the polynomial function divided by the leading coefficient.