If f is a polynomial with deg f = 3 and g is a polynomial with deg g = 2, what is the maximum possible degree of f(g(x))?

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Multiple Choice

If f is a polynomial with deg f = 3 and g is a polynomial with deg g = 2, what is the maximum possible degree of f(g(x))?

Explanation:
When a non-constant polynomial is composed with another non-constant polynomial, the degree multiplies: deg(f∘g) = deg f × deg g. Here deg f = 3 and deg g = 2, so the leading term of f(g(x)) comes from taking f’s cubic term and replacing its input with g(x), giving a·(g(x))^3. The highest-degree part of (g(x))^3 is (leading term of g)^3, which has degree 2×3 = 6. So deg(f(g(x))) is 6. This is the maximum possible degree, and it is actually achieved because there’s no cancellation of the top degree term.

When a non-constant polynomial is composed with another non-constant polynomial, the degree multiplies: deg(f∘g) = deg f × deg g. Here deg f = 3 and deg g = 2, so the leading term of f(g(x)) comes from taking f’s cubic term and replacing its input with g(x), giving a·(g(x))^3. The highest-degree part of (g(x))^3 is (leading term of g)^3, which has degree 2×3 = 6. So deg(f(g(x))) is 6. This is the maximum possible degree, and it is actually achieved because there’s no cancellation of the top degree term.

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