Overview of Polynomial Functions

Oct 13, 2025 by magicalceylontours.com 22 views

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Overview of Polynomial Functions

Polynomial functions are a fundamental concept in mathematics, and they play a crucial role in various fields, including algebra, calculus, and engineering. A polynomial function is a function that can be expressed as a sum of terms, where each term is a product of a constant and a variable raised to a non-negative integer power. In this article, we will focus on finding a third-degree polynomial function that satisfies certain conditions.

Third-Degree Polynomial Functions

A third-degree polynomial function is a polynomial function of degree three, which means that the highest power of the variable is three. The general form of a third-degree polynomial function is:

${ f(x) = ax^3 + bx^2 + cx + d }$

where ${ a }$ , ${ b }$ , ${ c }$ , and ${ d }$ are constants, and ${ x }$ is the variable.

Given Conditions

We are given that the polynomial function has an output of 1,025 when ${ x = 5 }$ , and has zeros ${ -20 }$ and ${ -4i }$ . This means that the function must satisfy the following conditions:

${ f(5) = 1025 }$
${ f(-20) = 0 }$
${ f(-4i) = 0 }$

Finding the Polynomial Function

To find the polynomial function, we can use the fact that if a polynomial function has a zero at a certain point, then the function can be factored as a product of linear factors. In this case, we know that the function has zeros at ${ -20 }$ and ${ -4i }$ , so we can write the function as:

${ f(x) = a(x + 20)(x + 4i) }$

Simplifying the Function

We can simplify the function by multiplying out the factors:

${ f(x) = a(x^2 + 24ix + 80) }$

Using the Given Condition

We are given that the function has an output of 1,025 when ${ x = 5 }$ , so we can substitute this value into the function to get:

${ f(5) = a(5^2 + 24i(5) + 80) }$

Solving for ${ a }$

We can simplify the expression and solve for ${ a }$ :

${ 1025 = a(25 + 120i + 80) }$

${ 1025 = a(105 + 120i) }$

${ a = \frac{1025}{105 + 120i} }$

Rationalizing the Denominator

To rationalize the denominator, we can multiply the numerator and denominator by the conjugate of the denominator:

${ a = \frac{1025}{105 + 120i} \times \frac{105 - 120i}{105 - 120i} }$

Simplifying the Expression

We can simplify the expression:

${ a = \frac{1025(105 - 120i)}{105^2 + 120^2} }$

${ a = \frac{1025(105 - 120i)}{11225 + 14400} }$

${ a = \frac{1025(105 - 120i)}{25625} }$

${ a = \frac{1085625 - 124000i}{25625} }$

${ a = 42.5 - 4.84i }$

Finding the Polynomial Function

Now that we have found the value of ${ a }$ , we can substitute it into the function:

${ f(x) = (42.5 - 4.84i)(x^2 + 24ix + 80) }$

Multiplying Out the Factors

We can multiply out the factors to get the final polynomial function:

${ f(x) = 42.5x^2 + 1172ix + 3400 - 384x - 121.92ix^2 - 240.96ix - 3840i }$

Simplifying the Expression

We can simplify the expression:

${ f(x) = (42.5 - 121.92i)x^2 + (1172i - 240.96ix - 384)x + (3400 - 3840i) }$

${ f(x) = (42.5 - 121.92i)x^2 + (1172i - 384 - 240.96ix)x + (3400 - 3840i) }$

${ f(x) = (42.5 - 121.92i)x^2 + (-384 - 240.96ix + 1172i)x + (3400 - 3840i) }$

${ f(x) = (42.5 - 121.92i)x^2 + (-384 + 1172i - 240.96ix)x + (3400 - 3840i) }$

${ f(x) = (42.5 - 121.92i)x^2 + (-384 + 1172i)x - 240.96ix + (3400 - 3840i) }$

${ f(x) = (42.5 - 121.92i)x^2 + (-384 + 1172i)x - 240.96ix + 3400 - 3840i }$

${ f(x) = (42.5 -

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Q: What is a third-degree polynomial function?

A: A third-degree polynomial function is a polynomial function of degree three, which means that the highest power of the variable is three. The general form of a third-degree polynomial function is:

[ f(x) = ax^3 + bx^2 + cx + d }$

Q: What are the given conditions for the polynomial function?

A: We are given that the polynomial function has an output of 1,025 when ${ x = 5 }$ , and has zeros ${ -20 }$ and ${ -4i }$ . This means that the function must satisfy the following conditions:

${ f(5) = 1025 }$
${ f(-20) = 0 }$
${ f(-4i) = 0 }$

Q: How do we find the polynomial function?

A: To find the polynomial function, we can use the fact that if a polynomial function has a zero at a certain point, then the function can be factored as a product of linear factors. In this case, we know that the function has zeros at ${ -20 }$ and ${ -4i }$ , so we can write the function as:

${ f(x) = a(x + 20)(x + 4i) }$

Q: How do we simplify the function?

A: We can simplify the function by multiplying out the factors:

${ f(x) = a(x^2 + 24ix + 80) }$

Q: How do we use the given condition to find the value of ${ a }$ ?

A: We are given that the function has an output of 1,025 when ${ x = 5 }$ , so we can substitute this value into the function to get:

${ f(5) = a(5^2 + 24i(5) + 80) }$

Q: How do we solve for ${ a }$ ?

A: We can simplify the expression and solve for ${ a }$ :

${ 1025 = a(25 + 120i + 80) }$

${ 1025 = a(105 + 120i) }$

${ a = \frac{1025}{105 + 120i} }$

Q: How do we rationalize the denominator?

A: To rationalize the denominator, we can multiply the numerator and denominator by the conjugate of the denominator:

${ a = \frac{1025}{105 + 120i} \times \frac{105 - 120i}{105 - 120i} }$

Q: How do we simplify the expression?

A: We can simplify the expression:

${ a = \frac{1025(105 - 120i)}{105^2 + 120^2} }$

${ a = \frac{1025(105 - 120i)}{11225 + 14400} }$

${ a = \frac{1025(105 - 120i)}{25625} }$

${ a = \frac{1085625 - 124000i}{25625} }$

${ a = 42.5 - 4.84i }$

Q: What is the final polynomial function?

A: Now that we have found the value of ${ a }$ , we can substitute it into the function:

${ f(x) = (42.5 - 4.84i)(x^2 + 24ix + 80) }$

Q: How do we multiply out the factors?

A: We can multiply out the factors to get the final polynomial function:

${ f(x) = 42.5x^2 + 1172ix + 3400 - 384x - 121.92ix^2 - 240.96ix - 3840i }$

Q: How do we simplify the expression?

A: We can simplify the expression:

${ f(x) = (42.5 - 121.92i)x^2 + (1172i - 384 - 240.96ix)x + (3400 - 3840i) }$

${ f(x) = (42.5 - 121.92i)x^2 + (-384 + 1172i - 240.96ix)x + (3400 - 3840i) }$

${ f(x) = (42.5 - 121.92i)x^2 + (-384 + 1172i)x - 240.96ix + (3400 - 3840i) }$

Q: What is the final answer?

A: The final polynomial function is:

${ f(x) = (42.5 - 121.92i)x^2 + (-384 + 1172i)x - 240.96ix + (3400 - 3840i) }$

Note: The final answer is a complex polynomial function, and it may not be possible to simplify it further.