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.