MHB Is the Degree of the Product of Two Polynomials 2n?

AI Thread Summary
The discussion centers on the degree of the product of two polynomials, specifically questioning whether it should be 2n or 2n-1. It is established that if A(x) and B(x) are both polynomials of degree n, their product C(x) should indeed have a degree of 2n, not 2n-1. The calculations for the coefficients of the product are presented, but there is confusion regarding the upper limit of the summation for the polynomial's degree. The conclusion reached is that the proposition stating the degree as 2n-1 is incorrect and should be revised to reflect a degree of 2n. The importance of verifying polynomial multiplication and checking results through examples is emphasized.
evinda
Gold Member
MHB
Messages
3,741
Reaction score
0
Hello! (Wave)

For polynomial multiplication, if $A(x)$ and $B(x)$ are polynomials of degree-bound $n$, we say that their product $C(x)$ is a polynomial of degree-bound $2n-1$ such that $C(x)=A(x)B(x)$ for all $x$ in the underlying field.
A way to express the product $C(x)$ is

$$C(x)= \sum_{j=0}^{2n-2} c_j x^j$$

where

$$c_j= \sum_{k=0}^j a_k b_{j-k}$$

Using the above equations, I want to find the product $A(x)B(x)$, where $A(x)=7x^3-x^2+x-10$, $B(x)=8x^3-6x+3$.

I found the following:

$c_0=-30 \\ c_1=63 \\ c_2=-89 \\ c_3=-53$

$c_4= a_0 b_4+a_1 b_3+a_2 b_2+ a_3 b_1+ a_4 b_0=8+21=29$

$c_5=a_0 b_5+ a_1 b_4+ a_2 b_3+ a_3 b_2+ a_4 b_1+ a_5 b_0=-8$

Is it right so far?

Also, is there a typo at this sum: $C(x)= \sum_{j=0}^{2n-2} c_j x^j$ ? Should the upper bound be $2n-1$ as at the above proposition is said?

So is the result $\sum_{k=0}^5 c_k x^k$ with the above calulated $c_k$? (Thinking)Thinking about it again, shouldn't the highest power of the product be $2n$?

If so, then $c_6=a_0 b_6+ a_1 b_5+ a_2 b_4+ a_3 b_3+ a_4 b_2+ a_5 b_1+ a_6 b_0=56$

So is it as follows?

$$C(x)=56 x^6-8x^5+29 x^4-53x^3-89x^2+63 x-30$$
 
Last edited:
Technology news on Phys.org
Come now evinda. What is the product of the two polynomials below (you do know how to multiply polynomials out by hand, yes?):
$$(7x^3−x^2+x−10)(8x^3−6x+3)$$
Once you have found the product manually, compare it with the solution you got, and with the solution you would have gotten following the proposition given. Can the proposition possibly be correct? Always check your work! If something doesn't look right, plug in an example to see if it comes out wrong!

Thinking about it again, shouldn't the highest power of the product be 2n?

What is the product of two linear polynomials (of degree 1)? Certainly it is a quadratic, a polynomial of degree 2. And 2 is twice 1, correct? The proposition concludes that the product of two linear polynomials has degree 0 (or 1, with 2n - 1) which is absurd. So it can't be right, and you are correct that it needs to be fixed to 2n.
 
Thread 'Is this public key encryption?'
I've tried to intuit public key encryption but never quite managed. But this seems to wrap it up in a bow. This seems to be a very elegant way of transmitting a message publicly that only the sender and receiver can decipher. Is this how PKE works? No, it cant be. In the above case, the requester knows the target's "secret" key - because they have his ID, and therefore knows his birthdate.
Dear Peeps I have posted a few questions about programing on this sectio of the PF forum. I want to ask you veterans how you folks learn program in assembly and about computer architecture for the x86 family. In addition to finish learning C, I am also reading the book From bits to Gates to C and Beyond. In the book, it uses the mini LC3 assembly language. I also have books on assembly programming and computer architecture. The few famous ones i have are Computer Organization and...
I tried a web search "the loss of programming ", and found an article saying that all aspects of writing, developing, and testing software programs will one day all be handled through artificial intelligence. One must wonder then, who is responsible. WHO is responsible for any problems, bugs, deficiencies, or whatever malfunctions which the programs make their users endure? Things may work wrong however the "wrong" happens. AI needs to fix the problems for the users. Any way to...
Back
Top