If $m(x)=x^2+3$ And $n(x)=5x+9$, Which Expression Is Equivalent To $ ( M ⋅ N ) ( X ) (m \cdot N)(x) ( M ⋅ N ) ( X ) [/tex]?A. 5 X 3 + 9 X 2 + 15 X + 27 5x^3 + 9x^2 + 15x + 27 5 X 3 + 9 X 2 + 15 X + 27 B. 25 X 2 + 90 X + 84 25x^2 + 90x + 84 25 X 2 + 90 X + 84 C. X 2 + 5 X + 12 X^2 + 5x + 12 X 2 + 5 X + 12 D. 5 X 2 + 24 5x^2 + 24 5 X 2 + 24

Understanding the Problem

To find the equivalent expression for $(m \cdot n)(x)$, we need to understand what the notation means. The notation $(m \cdot n)(x)$ represents the product of two functions, $m(x)$ and $n(x)$. In other words, we need to multiply the two functions together to get the resulting function.

Multiplying the Functions

To multiply the functions, we need to multiply each term in $m(x)$ by each term in $n(x)$. Let's start by multiplying the first term in $m(x)$, which is $x^2$, by each term in $n(x)$.

Multiplying $x^2$ by Each Term in $n(x)$

We have:

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

$$x^2 \cdot 9 = 9x^2$$

Multiplying the Constant Term in $m(x)$ by Each Term in $n(x)$

We have:

$$3 \cdot 5x = 15x$$

$$3 \cdot 9 = 27$$

Combining the Terms

Now, we need to combine the terms we got from multiplying $x^2$ by each term in $n(x)$ and the terms we got from multiplying the constant term in $m(x)$ by each term in $n(x)$.

We have:

$$(m \cdot n)(x) = 5x^3 + 9x^2 + 15x + 27$$

Evaluating the Options

Now, let's evaluate the options to see which one is equivalent to $(m \cdot n)(x)$.

Option A: $5x^3 + 9x^2 + 15x + 27$

Option B: $25x^2 + 90x + 84$

Option C: $x^2 + 5x + 12$

Option D: $5x^2 + 24$

Conclusion

Based on our calculation, we can see that the correct answer is:

Option A: $5x^3 + 9x^2 + 15x + 27$

This is the only option that matches the expression we got from multiplying the functions $m(x)$ and $n(x)$.

Why the Other Options are Incorrect

Let's take a look at the other options to see why they are incorrect.

Option B: $25x^2 + 90x + 84$

This option is incorrect because it is missing the term $5x^3$ and the term $9x^2$.

Option C: $x^2 + 5x + 12$

This option is incorrect because it is missing the terms $5x^3$, $9x^2$, and $15x$.

Option D: $5x^2 + 24$

This option is incorrect because it is missing the terms $5x^3$, $9x^2$, and $15x$.

Final Answer

The final answer is:

Option A: $5x^3 + 9x^2 + 15x + 27$

This is the only option that is equivalent to $(m \cdot n)(x)$.

Understanding Function Multiplication

In our previous article, we discussed how to multiply two functions together. We saw that the product of two functions, $m(x)$ and $n(x)$, is given by the expression $(m \cdot n)(x)$. In this article, we will answer some common questions about function multiplication.

Q: What is the difference between function addition and function multiplication?

A: Function addition and function multiplication are two different operations that are used to combine functions. Function addition is used to add two or more functions together, while function multiplication is used to multiply two or more functions together.

Q: How do I multiply two functions together?

A: To multiply two functions together, you need to multiply each term in one function by each term in the other function. For example, if we have two functions $m(x) = x^2 + 3$ and $n(x) = 5x + 9$, we can multiply them together as follows:

$$(m \cdot n)(x) = (x^2 + 3)(5x + 9)$$

Q: What is the order of operations for function multiplication?

A: The order of operations for function multiplication is the same as the order of operations for regular multiplication. You need to multiply the terms in the same order that they appear in the functions.

Q: Can I multiply a function by a constant?

A: Yes, you can multiply a function by a constant. For example, if we have a function $m(x) = x^2 + 3$ and we multiply it by a constant $5$, we get:

$$(5m)(x) = 5(x^2 + 3)$$

Q: Can I multiply two functions with different variables?

A: No, you cannot multiply two functions with different variables. For example, if we have two functions $m(x) = x^2 + 3$ and $n(y) = y^2 + 3$, we cannot multiply them together because they have different variables.

Q: What is the result of multiplying two functions with the same variable?

A: The result of multiplying two functions with the same variable is a new function that is the product of the two original functions. For example, if we have two functions $m(x) = x^2 + 3$ and $n(x) = 5x + 9$, we can multiply them together as follows:

$$(m \cdot n)(x) = (x^2 + 3)(5x + 9)$$

Q: Can I multiply a function by another function that is not a polynomial?

A: Yes, you can multiply a function by another function that is not a polynomial. For example, if we have a function $m(x) = x^2 + 3$ and we multiply it by a function $n(x) = \sin(x)$, we get:

$$(m \cdot n)(x) = (x^2 + 3)\sin(x)$$

Conclusion

In this article, we answered some common questions about function multiplication. We discussed the difference between function addition and function multiplication, how to multiply two functions together, and the order of operations for function multiplication. We also discussed what happens when you multiply a function by a constant or another function with the same variable.