Multiplying Polynomials: General Principles and Cauchy Products
Multiplying two arbitrary polynomials requires applying the generalized distributive property: every term of the first polynomial must be multiplied by every single term of the second polynomial.
The Convolution / Cauchy Formula
If $P(x) = \sum_{i=0}^n a_i x^i$ and $Q(x) = \sum_{j=0}^m b_j x^j$, their product $R(x) = P(x) \cdot Q(x)$ has degree $n + m$ with coefficients given by the discrete convolution:
c_k = \sum_{i=0}^k a_i b_{k-i}
For two quadratic polynomials $(a_2 x^2 + a_1 x + a_0)(b_2 x^2 + b_1 x + b_0)$:
- $c_4 = a_2 b_2$
- $c_3 = a_2 b_1 + a_1 b_2$
- $c_2 = a_2 b_0 + a_1 b_1 + a_0 b_2$
- $c_1 = a_1 b_0 + a_0 b_1$
- $c_0 = a_0 b_0$
Systematic Grid / Box Method
| $\times$ | $b_2 x^2$ | $b_1 x$ | $b_0$ |
|---|---|---|---|
| $a_2 x^2$ | $a_2 b_2 x^4$ | $a_2 b_1 x^3$ | $a_2 b_0 x^2$ |
| $a_1 x$ | $a_1 b_2 x^3$ | $a_1 b_1 x^2$ | $a_1 b_0 x$ |
| $a_0$ | $a_0 b_2 x^2$ | $a_0 b_1 x$ | $a_0 b_0$ |
Sum along the reverse diagonals to group identical powers of $x$.