Suppose that \(f\) is a polynomial and \(h\) is a polynomial of degree \(5\) or less such that \([(x^2+3x-3)\cdot f(x)+h(x)= -6x^6+72x^4 -50x^3 + 13x^2 +38x -44.\)
What is the degree of \(f\)
The degree of a polynomial is the highest power of x that occurs. The degree of the polynomial on the right hand side is therefore 6. That means the degree of the polynomial on the left hand side must also be 6. We are told that h(x) is only of degree 5, so the there must be a term in x6 that comes from multiplying f(x) by x2+3x-3. This means f(x) must have its largest power of x such that when multiplied by x2 it gives x6. Hence it must have a term with x4 as its largest power of x. That is, f is of degree 4.