We have that $3 \cdot f(x) + 4 \cdot g(x) = h(x)$ where $f(x),$ $g(x),$ and $h(x)$ are all polynomials in $x.$ If the degree of $f(x)$ is $8$ and the degree of $h(x)$ is $9$, then what is the minimum possible degree of $g(x)$?
If the degree of $h(x)$ is $9,$ that means there is a $x^9$ term in $h(x).$ That term cannot come from $f(x),$ since its degree is $8,$ so it must come from $g(x).$ That means the degree of $g(x)$ has to be at least $\boxed{9},$ and indeed, it can only be $9.$
9
Level 5
Algebra
Problem: We have that $3 \cdot f(x) + 4 \cdot g(x) = h(x)$ where $f(x),$ $g(x),$ and $h(x)$ are all polynomials in $x.$ If the degree of $f(x)$ is $8$ and the degree of $h(x)$ is $9$, then what is the minimum possible degree of $g(x)$?
Solution:<|START_OF_LATENT|><|PRIOR_PREFIX|>To determine the minimum possible degree of \( g(x) \), we start by analyzing the equation \( 3 \cdot f(x) + 4 \cdot g(x) = h(x) \). The degree of a polynomial is defined as the highest power of the variable in the polynomial. 

Given that the degree of \( f(x) \) is \( 8 \), we can express this as \( \deg(f) = 8 \). The term \( 3 \cdot f(x) \) will also have a degree of \( 8 \) since multiplying a polynomial by a non-zero constant does not change its degree. Thus, we have \( \deg(3 \cdot f(x)) = 8 \).

Next, we know that \( h(x) \) has a degree of \( 9 \), expressed as \( \deg(h) = 9 \). The degree of the sum of two polynomials is determined by the polynomial with the highest degree, unless the leading terms cancel each other out. In this case, since \( \deg(h) = 9 \) and \( \deg(3 \cdot f(x)) = 8 \), for the equation to hold true, \( g(x) \) must contribute a degree that allows the overall degree of the left-hand side to equal \( 9 \).

Let \( \deg(g) = d \). The term \( 4 \cdot g(x) \) will have a degree of \( d \). For the equation \( 3 \cdot f(x) + 4 \cdot g(x) = h(x) \) to yield a polynomial \( h(x) \) of degree \( 9 \), the degree \( d \) must be at least \( 9 \) because \( \deg(h) \) is greater than \( \deg(3 \cdot f(x)) \). 

If \( d < 9 \), then \( \deg(3 \cdot f(x) + 4 \cdot g(x)) \) would be \( 8 \), which contradicts \( \deg(h) = 9 \). Therefore, the minimum degree \( d \) must be \( 9 \) to ensure that the left-hand side can reach the degree of \( 9 \) required for \( h(x) \).

Thus, the minimum possible degree of \( g(x) \) is \( 9 \).<|END_OF_LATENT|> The final answer is $9$.