Write A Polynomial Function Of Least Degree With Integral Coefficients That Has The Given Zeros: \[$-3, 1, -3i\$\].A. \[$f(x) = X^4 + 2x^3 + 5x^2 + 8x + 4\$\]B. \[$f(x) = X^4 - 3x^3 - 9x^2 + 77x + 150\$\]C. \[$f(x) = X^4 +

Understanding the Problem

When given the zeros of a polynomial function, we can use this information to write the function in factored form. In this case, we are given the zeros: $-3, 1, -3i$. Our goal is to write a polynomial function of least degree with integral coefficients that has these given zeros.

The Concept of Zeros in Polynomial Functions

In mathematics, the zeros of a polynomial function are the values of $x$ that make the function equal to zero. In other words, if we have a polynomial function $f(x)$, then the zeros of $f(x)$ are the values of $x$ such that $f(x) = 0$. The given zeros $-3, 1, -3i$ are the values of $x$ that make the polynomial function equal to zero.

Writing the Polynomial Function in Factored Form

To write the polynomial function in factored form, we can use the fact that if $a$ is a zero of the polynomial function $f(x)$, then $(x - a)$ is a factor of $f(x)$. In this case, we have three zeros: $-3, 1, -3i$. Therefore, we can write the polynomial function in factored form as:

$$f(x) = (x + 3)(x - 1)(x + 3i)(x - 3i)$$

Multiplying the Factors

To find the polynomial function in standard form, we need to multiply the factors together. We can do this by multiplying the first two factors, and then multiplying the result by the last two factors.

$$(x + 3)(x - 1) = x^2 + 2x - 3$$

$$(x^2 + 2x - 3)(x + 3i)(x - 3i) = (x^2 + 2x - 3)(x^2 + 9)$$

$$(x^2 + 2x - 3)(x^2 + 9) = x^4 + 9x^2 + 2x^3 + 18x - 3x^2 - 27$$

$$(x^4 + 9x^2 + 2x^3 + 18x - 3x^2 - 27) = x^4 + 6x^2 + 2x^3 + 18x - 27$$

Simplifying the Expression

We can simplify the expression by combining like terms.

$$x^4 + 6x^2 + 2x^3 + 18x - 27 = x^4 + 2x^3 + 6x^2 + 18x - 27$$

Conclusion

In conclusion, we have written a polynomial function of least degree with integral coefficients that has the given zeros: $-3, 1, -3i$. The polynomial function is:

$$f(x) = x^4 + 2x^3 + 6x^2 + 18x - 27$$

This polynomial function has the given zeros and is of least degree with integral coefficients.

Comparison with the Given Options

Let's compare our polynomial function with the given options:

A. $f(x) = x^4 + 2x^3 + 5x^2 + 8x + 4$

B. $f(x) = x^4 - 3x^3 - 9x^2 + 77x + 150$

C. $f(x) = x^4 + 2x^3 + 5x^2 + 8x + 4$

Our polynomial function is different from the given options. However, we can see that option A is close to our polynomial function, but it has a different constant term.

Final Answer

The final answer is:

f(x) = x^4 + 2x^3 + 6x^2 + 18x - 27$<br/> # **Frequently Asked Questions about Writing Polynomial Functions** ## **Q: What is the difference between a zero and a root of a polynomial function?** A: In mathematics, the terms "zero" and "root" are often used interchangeably to refer to the values of $x$ that make a polynomial function equal to zero. However, some mathematicians make a distinction between the two terms. A zero is a value of $x$ that makes the polynomial function equal to zero, while a root is a value of $x$ that makes the polynomial function equal to zero, and is also a solution to the equation. ## **Q: How do I find the polynomial function in standard form?** A: To find the polynomial function in standard form, you need to multiply the factors together. You can do this by multiplying the first two factors, and then multiplying the result by the last two factors. ## **Q: What is the significance of the degree of a polynomial function?** A: The degree of a polynomial function is the highest power of $x$ in the polynomial function. For example, in the polynomial function $f(x) = x^4 + 2x^3 + 6x^2 + 18x - 27$, the degree is 4. The degree of a polynomial function is important because it determines the number of zeros the polynomial function has. ## **Q: Can a polynomial function have more than one zero?** A: Yes, a polynomial function can have more than one zero. In fact, a polynomial function can have any number of zeros, including zero, one, two, three, or more. ## **Q: How do I determine the number of zeros a polynomial function has?** A: To determine the number of zeros a polynomial function has, you need to look at the degree of the polynomial function. If the degree is even, then the polynomial function has an even number of zeros. If the degree is odd, then the polynomial function has an odd number of zeros. ## **Q: Can a polynomial function have complex zeros?** A: Yes, a polynomial function can have complex zeros. In fact, complex zeros are a common feature of polynomial functions. ## **Q: How do I write a polynomial function in factored form?** A: To write a polynomial function in factored form, you need to identify the zeros of the polynomial function and then write the polynomial function as a product of factors, where each factor is of the form $(x - a)$, where $a$ is a zero of the polynomial function. ## **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 polynomial function equal to zero. The factors of a polynomial function are the expressions that, when multiplied together, give the polynomial function. The zeros of a polynomial function are the values of $x$ that make each factor equal to zero. ## **Q: Can a polynomial function have a zero that is not a rational number?** A: Yes, a polynomial function can have a zero that is not a rational number. In fact, complex zeros are a common feature of polynomial functions. ## **Q: How do I determine if a polynomial function has a rational zero?** A: To determine if a polynomial function has a rational zero, you need to look at the factors of the polynomial function. If a factor is of the form $(x - a)$, where $a$ is a rational number, then the polynomial function has a rational zero. ## **Q: What is the significance of the constant term of a polynomial function?** A: The constant term of a polynomial function is the term that is not multiplied by any power of $x$. The constant term is important because it determines the value of the polynomial function when $x$ is equal to zero. ## **Q: Can a polynomial function have a constant term that is not an integer?** A: Yes, a polynomial function can have a constant term that is not an integer. In fact, the constant term can be any real number. ## **Q: How do I determine the degree of a polynomial function?** A: To determine the degree of a polynomial function, you need to look at the highest power of $x$ in the polynomial function. The degree of a polynomial function is the highest power of $x$ in the polynomial function. ## **Q: What is the relationship between the degree of a polynomial function and its zeros?** A: The degree of a polynomial function is related to the number of zeros the polynomial function has. If the degree is even, then the polynomial function has an even number of zeros. If the degree is odd, then the polynomial function has an odd number of zeros. ## **Q: Can a polynomial function have a zero that is a perfect square?** A: Yes, a polynomial function can have a zero that is a perfect square. In fact, perfect squares are a common feature of polynomial functions. ## **Q: How do I determine if a polynomial function has a zero that is a perfect square?** A: To determine if a polynomial function has a zero that is a perfect square, you need to look at the factors of the polynomial function. If a factor is of the form $(x - a)$, where $a$ is a perfect square, then the polynomial function has a zero that is a perfect square. ## **Q: What is the significance of the leading coefficient of a polynomial function?** A: The leading coefficient of a polynomial function is the coefficient of the highest power of $x$ in the polynomial function. The leading coefficient is important because it determines the value of the polynomial function when $x$ is equal to zero. ## **Q: Can a polynomial function have a leading coefficient that is not an integer?** A: Yes, a polynomial function can have a leading coefficient that is not an integer. In fact, the leading coefficient can be any real number. ## **Q: How do I determine the leading coefficient of a polynomial function?** A: To determine the leading coefficient of a polynomial function, you need to look at the coefficient of the highest power of $x$ in the polynomial function. ## **Q: What is the relationship between the leading coefficient of a polynomial function and its zeros?** A: The leading coefficient of a polynomial function is related to the number of zeros the polynomial function has. If the leading coefficient is positive, then the polynomial function has an even number of zeros. If the leading coefficient is negative, then the polynomial function has an odd number of zeros. ## **Q: Can a polynomial function have a leading coefficient that is a perfect square?** A: Yes, a polynomial function can have a leading coefficient that is a perfect square. In fact, perfect squares are a common feature of polynomial functions. ## **Q: How do I determine if a polynomial function has a leading coefficient that is a perfect square?** A: To determine if a polynomial function has a leading coefficient that is a perfect square, you need to look at the factors of the polynomial function. If a factor is of the form $(x - a)$, where $a$ is a perfect square, then the polynomial function has a leading coefficient that is a perfect square.