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It looks like you are working on polynomial operations with the functions \( f(x) = 5x^5 + 10x^2 \) and \( g(x) = 2x^5 - 10x^3 + 9 \). You are asked to evaluate the following polynomial operations:

1. \( f(x) + g(x) \)
2. \( f(x) - g(x) \)
3. \( f(x) \cdot g(x) \)

I'll guide you through each operation:

### 1. \( f(x) + g(x) \)

We add the corresponding terms of the polynomials \( f(x) \) and \( g(x) \):

\[
f(x) + g(x) = (5x^5 + 10x^2) + (2x^5 - 10x^3 + 9)
\]
\[
= (5x^5 + 2x^5) + (-10x^3) + (10x^2) + 9
\]
\[
= 7x^5 - 10x^3 + 10x^2 + 9
\]

### 2. \( f(x) - g(x) \)

We subtract the terms of \( g(x) \) from \( f(x) \):

\[
f(x) - g(x) = (5x^5 + 10x^2) - (2x^5 - 10x^3 + 9)
\]
\[
= (5x^5 - 2x^5) + (-(-10x^3)) + (10x^2 - 0) - 9
\]
\[
= 3x^5 + 10x^3 + 10x^2 - 9
\]

### 3. \( f(x) \cdot g(x) \)

For multiplication, we apply the distributive property (FOIL method), multiplying each term of \( f(x) \) with each term of \( g(x) \):

\[
f(x) \cdot g(x) = (5x^5 + 10x^2)(2x^5 - 10x^3 + 9)
\]
Expanding this step by step:

First, multiply \( 5x^5 \):
\[
5x^5 \cdot 2x^5 = 10x^{10}
\]
\[
5x^5 \cdot (-10x^3) = -50x^8
\]
\[
5x^5 \cdot 9 = 45x^5
\]

Now, multiply \( 10x^2 \):
\[
10x^2 \cdot 2x^5 = 20x^7
\]
\[
10x^2 \cdot (-10x^3) = -100x^5
\]
\[
10x^2 \cdot 9 = 90x^2
\]

Now combine the terms:
\[
f(x) \cdot g(x) = 10x^{10} - 50x^8 + 20x^7 + (45x^5 - 100x^5) + 90x^2
\]
\[
= 10x^{10} - 50x^8 + 20x^7 - 55x^5 + 90x^2
\]

These are the correct operations for the polynomials.

Let me know if you need more details or if you have any questions!

### Relative Questions:
1. What happens if we substitute a specific value for \( x \) in these polynomial expressions?
2. Can you explain how the degree of a polynomial changes during addition, subtraction, and multiplication?
3. How would you approach finding the roots of the resulting polynomials?
4. What are some real-world applications where polynomial multiplication is useful?
5. How can we simplify polynomial expressions with higher powers?

### Tip:
When adding or subtracting polynomials, always align terms with the same powers of \( x \) to avoid mistakes in combining them.

Given the functions f(x) = 5x^5 + 10x^2 and g(x) = 2x^5 - 10x^3 + 9, determine each of the following. Give your answers as simplified expressions written in descending order. Evaluate f + g, f - g, and f * g.

Learn how to perform polynomial operations including addition, subtraction, and multiplication with the functions f(x) = 5x^5 + 10x^2 and g(x) = 2x^5 - 10x^3 + 9. This guide covers step-by-step solutions, combining like terms, and simplifying expressions.

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