jmc

Question

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)?

Accepted Answer

Let f(x) = a_3 x^3 + a_2 x^2 + a_1 x + a_0 and g(x) = b_5 x^5 + b_4 x^4 + b_3 x^3 + b_2 x^2 + b_1 x + b_0. Then align* 2f(x) + 4g(x) &= 2 (a_3 x^3 + a_2 x^2 + a_1 x + a_0) & + 4 (b_5 x^5 + b_4 x^4 + b_3 x^3 + b_2 x^2 + b_1 x + b_0) &= 4b_5 x^5 + 4b_4 x^4 + (2a_3 + 4b_3) x^3 + (2a_2 + 4b_2) x^2 & + (2a_1 + 4b_1) x + (2a_0 + 4b_0). align*Thus, the degree of 2f(x) + 4g(x) is 5.