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.
Isolating $p(x),$ we have: \begin{align*}
p(x)&=(7x^3+24x^2+25x+1)-(x^5+3x^3+9x)\\
&=-x^5+(7-3)x^3+24x^2+(25-9)x+1\\
&=\boxed{-x^5+4x^3+24x^2+16x+1}.
\end{align*}
-x^5+4x^3+24x^2+16x+1
Level 4
Algebra
Problem: 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.
Solution:<|START_OF_LATENT|><|PRIOR_PREFIX|>To find the polynomial $p(x)$, we start with the equation given: 

$$p(x) + (x^5 + 3x^3 + 9x) = (7x^3 + 24x^2 + 25x + 1).$$ 

We need to isolate $p(x)$. This can be done by subtracting the polynomial $(x^5 + 3x^3 + 9x)$ from both sides of the equation. 

Thus, we rewrite the equation as:

$$p(x) = (7x^3 + 24x^2 + 25x + 1) - (x^5 + 3x^3 + 9x).$$ 

Next, we distribute the negative sign across the terms in the second polynomial:

$$p(x) = 7x^3 + 24x^2 + 25x + 1 - x^5 - 3x^3 - 9x.$$

Now, we combine like terms. We start with the highest degree term:

1. The $x^5$ term: There is only $-x^5$, so we have $-x^5$.
2. The $x^3$ terms: We have $7x^3 - 3x^3 = 4x^3$.
3. The $x^2$ term: There is only $24x^2$, so we have $24x^2$.
4. The $x$ terms: We have $25x - 9x = 16x$.
5. The constant term: There is only $1$, so we have $1$.

Putting all these together, we get:

$$p(x) = -x^5 + 4x^3 + 24x^2 + 16x + 1.$$

This expression is a polynomial with the degrees of the terms arranged in decreasing order, confirming that the final answer is $-x^5 + 4x^3 + 24x^2 + 16x + 1$.<|END_OF_LATENT|> The final answer is $-x^5+4x^3+24x^2+16x+1$.