Polynomials of different degrees and a related monomial

[saadsarfraz]

Homework Statement

Let f, g be nonzero polynomials with deg (f) $$\geq$$ deg (g). Show that there
is a unique monomial bx$$^{k}$$ where deg(f(x) - bx$$^{k}$$g(x)) < deg (f).

Homework Equations

see above

The Attempt at a Solution

I define polynomials f and g, with deg(f) = n and deg (g) = m and n$$\geq$$m
and let the monomial be h(x) so h(x)g(x) = l(x) and using the theorem deg(h(x)g(x)) = deg(h(x)+g(x)) and therefore deg l(x) = k + m. so overall i have to find deg(f(x)-l(x)) but this is equal to f(x) not less than f(x), how do I show its less than f(x).

 

[epenguin]

In other words you have a polynomial f = a_nxⁿ + a_n-1x^n-1 + ... , and you have another one g = c_mx^m + c_m-1x^m-1 + ...
where n > m .

And you are asked if you can find a bx^k that can make the degree of
(f - bx^kg) less than n.

In other words the coefficient of xⁿ in the polynomial (f - bx^kg) has to be what?

 

[saadsarfraz]

the coefficient has to be 0 than I think? but degree k+m still could be greater than n as we don't know anything about k.

 

[epenguin]

You are just asked can you find, will you always be able to find, a k (and a b) that gives you the result you want?

Maybe you would find it easier if you first considered a concrete case. I can choose any polynomials that come into my head as long as n >or= m.

f = 5x³ + 10x² + 2x + 8.5 and g = 3x + 2 comes into my head.

Can you find a b and k for that that gives you a reduced degree result for (f - bx^kg) ?