Find A 3rd-degree Polynomial Function With Real Coefficients, A Leading Coefficient Of 1, And Satisfying The Given Conditions.Conditions: - ${ 5\$} And ${ 3i\$} Are Zeros. { F(x) = \square$}$(Simplify Your Answer.)

Introduction

In algebra, a polynomial function is a function that can be written in the form of a sum of terms, where each term is a product of a constant and a variable raised to a non-negative integer power. A 3rd-degree polynomial function is a polynomial function of degree 3, which means the highest power of the variable is 3. In this article, we will find a 3rd-degree polynomial function with real coefficients, a leading coefficient of 1, and satisfying the given conditions.

Conditions

The given conditions are that the polynomial function has real coefficients, a leading coefficient of 1, and has zeros at $5$ and $3i$. Since the coefficients are real, the complex zeros must come in conjugate pairs. Therefore, the other zero is $-3i$.

Finding the Polynomial Function

To find the polynomial function, we can use the fact that if $r$ is a zero of a polynomial function, then $(x-r)$ is a factor of the polynomial function. Since the zeros are $5$ and $-3i$, we know that $(x-5)$ and $(x+3i)$ are factors of the polynomial function.

Complex Conjugate Roots Theorem

The Complex Conjugate Roots Theorem states that if a polynomial function has real coefficients and a complex zero, then the complex conjugate of the zero is also a zero of the polynomial function. Therefore, since $-3i$ is a zero, $3i$ is also a zero.

Writing the Polynomial Function

Since the leading coefficient is 1, the polynomial function can be written in the form of:

$$f(x) = (x-5)(x+3i)(x-(-3i))$$

Simplifying the Polynomial Function

To simplify the polynomial function, we can multiply the factors together:

$$f(x) = (x-5)(x+3i)(x+3i)$$

$$f(x) = (x-5)(x^2+6ix-9)$$

$$f(x) = x^3+6ix^2-9x-5x^2-30i-45$$

$$f(x) = x^3+(6i-5)x^2-9x-30i-45$$

Combining Like Terms

To combine like terms, we can group the terms with the same power of $x$ together:

$$f(x) = x^3+(-5+6i)x^2-9x-30i-45$$

Writing the Final Answer

The final answer is a 3rd-degree polynomial function with real coefficients, a leading coefficient of 1, and satisfying the given conditions:

$$f(x) = x^3+(-5+6i)x^2-9x-30i-45$$

Conclusion

In this article, we found a 3rd-degree polynomial function with real coefficients, a leading coefficient of 1, and satisfying the given conditions. The polynomial function has zeros at $5$ and $3i$, and the complex conjugate of $3i$ is $-3i$. We used the Complex Conjugate Roots Theorem to find the other zero, and then we multiplied the factors together to simplify the polynomial function.

Discussion

The discussion category for this article is mathematics. The article is about finding a 3rd-degree polynomial function with real coefficients, a leading coefficient of 1, and satisfying the given conditions. The article uses the Complex Conjugate Roots Theorem to find the other zero, and then it multiplies the factors together to simplify the polynomial function.

Related Topics

• Complex Conjugate Roots Theorem
• Polynomial Functions
• Algebra
• Mathematics

References

• Complex Conjugate Roots Theorem
• Polynomial Functions
• Algebra
• Mathematics

Tags

• Complex Conjugate Roots Theorem
• Polynomial Functions
• Algebra
• Mathematics
• 3rd-degree polynomial function
• Real coefficients
• Leading coefficient of 1
• Zeros at $5$ and $3i$
• Complex conjugate of $3i$ is $-3i$
  

Introduction

In our previous article, we found a 3rd-degree polynomial function with real coefficients, a leading coefficient of 1, and satisfying the given conditions. In this article, we will answer some frequently asked questions related to the topic.

Q&A

Q1: What is a 3rd-degree polynomial function?

A1: A 3rd-degree polynomial function is a polynomial function of degree 3, which means the highest power of the variable is 3.

Q2: What are the conditions for the polynomial function?

A2: The conditions for the polynomial function are that it has real coefficients, a leading coefficient of 1, and has zeros at $5$ and $3i$.

Q3: Why do we need to find the complex conjugate of $3i$?

A3: We need to find the complex conjugate of $3i$ because the Complex Conjugate Roots Theorem states that if a polynomial function has real coefficients and a complex zero, then the complex conjugate of the zero is also a zero of the polynomial function.

Q4: How do we simplify the polynomial function?

A4: We simplify the polynomial function by multiplying the factors together.

Q5: What is the final answer?

A5: The final answer is a 3rd-degree polynomial function with real coefficients, a leading coefficient of 1, and satisfying the given conditions:

$$f(x) = x^3+(-5+6i)x^2-9x-30i-45$$

Q6: What are some related topics?

A6: Some related topics are Complex Conjugate Roots Theorem, Polynomial Functions, Algebra, and Mathematics.

Q7: What are some references for further reading?

A7: Some references for further reading are Complex Conjugate Roots Theorem, Polynomial Functions, Algebra, and Mathematics.

Q8: What are some tags for this article?

A8: Some tags for this article are Complex Conjugate Roots Theorem, Polynomial Functions, Algebra, Mathematics, 3rd-degree polynomial function, Real coefficients, Leading coefficient of 1, Zeros at $5$ and $3i$, Complex conjugate of $3i$ is $-3i$.

Conclusion

In this article, we answered some frequently asked questions related to finding a 3rd-degree polynomial function with real coefficients, a leading coefficient of 1, and satisfying the given conditions. We hope that this article has been helpful in clarifying any doubts that you may have had.

Discussion

The discussion category for this article is mathematics. The article is about finding a 3rd-degree polynomial function with real coefficients, a leading coefficient of 1, and satisfying the given conditions. The article uses the Complex Conjugate Roots Theorem to find the other zero, and then it multiplies the factors together to simplify the polynomial function.

Related Topics

• Complex Conjugate Roots Theorem
• Polynomial Functions
• Algebra
• Mathematics

References

• Complex Conjugate Roots Theorem
• Polynomial Functions
• Algebra
• Mathematics

Tags

• Complex Conjugate Roots Theorem
• Polynomial Functions
• Algebra
• Mathematics
• 3rd-degree polynomial function
• Real coefficients
• Leading coefficient of 1
• Zeros at $5$ and $3i$
• Complex conjugate of $3i$ is $-3i$