### Instruction: If $f(x)$ is a polynomial of degree 3, and $g(x)$ is a polynomial of degree 5, then what is the degree of polynomial $2f(x) + 4g(x)$?### Response: To find the degree of a polynomial, I need to look at the highest power of $x$ that appears in the polynomial.
If I multiply a polynomial by a constant, like 2 or 4, that does not change the degree of the polynomial, because it just scales the coefficients.
So, the degree of $2f(x)$ is the same as the degree of $f(x)$, which is 3, and the degree of $4g(x)$ is the same as the degree of $g(x)$, which is 5.
To find the degree of a sum of two polynomials, I need to look at the terms with the highest powers of $x$ in each polynomial, and see which one is larger.
In this case, the term with the highest power of $x$ in $2f(x)$ is $2a_3x^3$, where $a_3$ is the coefficient of $x^3$ in $f(x)$, and the term with the highest power of $x$ in $4g(x)$ is $4b_5x^5$, where $b_5$ is the coefficient of $x^5$ in $g(x)$.
Since $x^5$ is a higher power than $x^3$, the degree of $2f(x) + 4g(x)$ is determined by the degree of $4g(x)$, which is 5.