Select The Correct Answer From Each Drop-down Menu.If $f(x) = 0.5x^2 - 2$ And $g(x) = 8x^3 + 2$, Find The Value Of The Following Function:$\[ (f \cdot G)(x) = \square \, X^5 - \square \, X^3 + \square \, X^2 - \square \\]

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Introduction

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In mathematics, polynomial functions are a fundamental concept in algebra and calculus. When we multiply two polynomial functions, we need to apply the distributive property to each term in the first function and multiply it by each term in the second function. In this article, we will explore how to multiply two polynomial functions and find the value of the resulting function.

Understanding Polynomial Functions

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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 $a_n, a_{n-1}, \ldots, a_1, a_0$ are constants, and $n$ is a non-negative integer.

In this article, we will be working with two polynomial functions:

$$f(x) = 0.5x^2 - 2$$

$$g(x) = 8x^3 + 2$$

Multiplying Two Polynomial Functions

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To multiply two polynomial functions, we need to apply the distributive property to each term in the first function and multiply it by each term in the second function.

Let's start by multiplying the first term in $f(x)$, which is $0.5x^2$, by each term in $g(x)$:

$$0.5x^2 \cdot 8x^3 = 4x^5$$

$$0.5x^2 \cdot 2 = x^2$$

Now, let's multiply the second term in $f(x)$, which is $-2$, by each term in $g(x)$:

$$-2 \cdot 8x^3 = -16x^3$$

$$-2 \cdot 2 = -4$$

Combining Like Terms

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Now that we have multiplied each term in $f(x)$ by each term in $g(x)$, we need to combine like terms. Like terms are terms that have the same variable and exponent.

In this case, we have two like terms: $4x^5$ and $-16x^5$. We can combine these terms by adding their coefficients:

$$4x^5 - 16x^5 = -12x^5$$

We also have two like terms: $x^2$ and $-4$. We can combine these terms by adding their coefficients:

$$x^2 - 4$$

The Final Result

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Now that we have combined like terms, we can write the final result:

$$(f \cdot g)(x) = -12x^5 - 16x^3 + x^2 - 4$$

Conclusion

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In this article, we have explored how to multiply two polynomial functions and find the value of the resulting function. We have applied the distributive property to each term in the first function and multiplied it by each term in the second function. We have then combined like terms to simplify the resulting expression.

By following these steps, you can multiply two polynomial functions and find the value of the resulting function.

Example Problems

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Problem 1

Find the value of the following function:

$$(f \cdot g)(x) = \square \, x^5 - \square \, x^3 + \square \, x^2 - \square$$

where $f(x) = 2x^2 + 1$ and $g(x) = 3x^3 - 2$.

Solution

To solve this problem, we need to multiply each term in $f(x)$ by each term in $g(x)$ and combine like terms.

Let's start by multiplying the first term in $f(x)$, which is $2x^2$, by each term in $g(x)$:

$$2x^2 \cdot 3x^3 = 6x^5$$

$$2x^2 \cdot -2 = -4x^2$$

Now, let's multiply the second term in $f(x)$, which is $1$, by each term in $g(x)$:

$$1 \cdot 3x^3 = 3x^3$$

$$1 \cdot -2 = -2$$

Now, let's combine like terms:

$$6x^5 - 4x^2 + 3x^3 - 2$$

Problem 2

Find the value of the following function:

$$(f \cdot g)(x) = \square \, x^5 - \square \, x^3 + \square \, x^2 - \square$$

where $f(x) = 4x^2 - 1$ and $g(x) = 2x^3 + 3$.

Solution

To solve this problem, we need to multiply each term in $f(x)$ by each term in $g(x)$ and combine like terms.

Let's start by multiplying the first term in $f(x)$, which is $4x^2$, by each term in $g(x)$:

$$4x^2 \cdot 2x^3 = 8x^5$$

$$4x^2 \cdot 3 = 12x^2$$

Now, let's multiply the second term in $f(x)$, which is $-1$, by each term in $g(x)$:

$$-1 \cdot 2x^3 = -2x^3$$

$$-1 \cdot 3 = -3$$

Now, let's combine like terms:

$$8x^5 + 12x^2 - 2x^3 - 3$$

Final Answer

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The final answer is: $\boxed{-12x^5 - 16x^3 + x^2 - 4}$