evinda
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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?
$$C(x)=56 x^6-8x^5+29 x^4-53x^3-89x^2+63 x-30$$
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?
$$C(x)=56 x^6-8x^5+29 x^4-53x^3-89x^2+63 x-30$$
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