### Instruction: Suppose $p(x)$ is a function such that $p(x) + (x^5+3x^3+9x) = (7x^3+24x^2+25x+1)$.  Express $p(x)$ as a polynomial with the degrees of the terms in decreasing order.### Response: I want to find a function $p(x)$ that satisfies the given equation, so I'll start by rearranging the equation to isolate $p(x)$ on one side.
To do that, I'll subtract $(x^5+3x^3+9x)$ from both sides, which gives me $p(x) = (7x^3+24x^2+25x+1) - (x^5+3x^3+9x)$.
Now I have $p(x)$ as the difference of two polynomials, so I can simplify it by combining like terms.
The only term that has a degree of $5$ is $-x^5$, so that's the first term of $p(x)$.
Then I look at the terms that have a degree of $3$.
I have $7x^3$ and $-3x^3$, so I can subtract them and get $4x^3$.
Next I look at the terms that have a degree of $2$.
I only have $24x^2$, so that's the next term of $p(x)$.
Then I look at the terms that have a degree of $1$.
I have $25x$ and $-9x$, so I can subtract them and get $16x$.
Finally, I look at the constant terms.
I only have $1$, so that's the last term of $p(x)$.
So I have found $p(x)$ as a polynomial with the degrees of the terms in decreasing order.