# Thread: Prove of the Lemma...

1. ## Prove of the Lemma...

Suppose p(x), q(x) and r(x) E F(x), where F(x) can be represented as complex, Real or Rational polynomials...

To Prove:

1) If neither p(x) nor q(x) is the zero polynomial, then
deg(p(x) q(x)) = deg(p(x)) + deg (q(x)).

2) If r(x) is not the zero polynomial and r(x).p(x) = r(x).q(x), then
p(x) = q(x)

2. Originally Posted by vikramtiwari
Suppose p(x), q(x) and r(x) E F(x), where F(x) can be represented as complex, Real or Rational polynomials...

To Prove:

1) If neither p(x) nor q(x) is the zero polynomial, then
deg(p(x) q(x)) = deg(p(x)) + deg (q(x)).

2) If r(x) is not the zero polynomial and r(x).p(x) = r(x).q(x), then
p(x) = q(x)
How is degree defined to you?
I will assume that the degree of the polynomial is the number of complex roots.And all the polynomials mentioned have finite degree.

1) Count the number of roots. for p(x)q(x) = 0, roots of any of the polynomial p(x) or q(x) will do. So there are totally deg(p(x)) + deg (q(x)) roots. Thus the answer.

2)If for all values of x, we have r(x).p(x) = r(x).q(x), then for all values of x, r(x).[p(x)-q(x)] = 0. This forces [p(x)-q(x)] to be identically zero (since r(x) is non zero). Thus p(x) = q(x).

P.S: I think TPH and Mathguru had a similar discussion in the forum somewhere

3. I think by degree they mean the sum of the highest power...

that is for example, the polynomial 7x2y3 + 4x − 9 has three terms, where the polynomial can also be expressed as 7x2y3 + 4x1y0 − 9x0y0.) The first term has a degree of 5 (the sum of 2 and 3), the second term has a degree of 1, and the last term has a degree of 0. Therefore, the polynomial has a degree of 5 which is the highest degree of any term.

Since, (y − 3)(2y + 6)( − 4y − 21) = − 8y3 − 42y2 + 72y + 378, the degree of the polynomial is 3.

So, will that make any change to what you have explained it above...

Thanks!

P.S. What is TPH? and do we have to pay and register for Mathguru?

4. Originally Posted by vikramtiwari
I think by degree they mean the sum of the highest power...

that is for example, the polynomial 7x2y3 + 4x − 9 has three terms, where the polynomial can also be expressed as 7x2y3 + 4x1y0 − 9x0y0.) The first term has a degree of 5 (the sum of 2 and 3), the second term has a degree of 1, and the last term has a degree of 0. Therefore, the polynomial has a degree of 5 which is the highest degree of any term.

Since, (y − 3)(2y + 6)( − 4y − 21) = − 8y3 − 42y2 + 72y + 378, the degree of the polynomial is 3.

So, will that make any change to what you have explained it above...

Thanks!
I just tried to do it differently. With your definition, (1) will need multiplying the terms out.
Just write the definitions:
$p(x) = \sum_{k=0}^{k= \, deg \, (p(x))} p_k x^k$ and $q(x) = \sum_{k=0}^{k= \, deg \, (q(x))} q_k x^k$
So:
$p(x)q(x) = \left(\sum_{k=0}^{k= \, deg \, (p(x))} p_k x^k\right) \cdot \left(\sum_{k=0}^{k= \, deg \, (q(x))} q_k x^k\right)$

$= \sum_{k=0}^{k= deg(q(x)) + deg(p(x))} \left(\sum_{j=0}^{j=k} p_j q_{k-j}\right) x^k$

P.S. What is TPH? and do we have to pay and register for Mathguru?
TPH and Mathguru are user ids of 2 people on this site :P
One of them is a moderator, the other is an admin