多项式运算

0. 表示

• x4−3x+1$x^4-3x+1$ ⇒ {(1,4),(−3,1),(1,0)}$\{\left(1,4\right),\left(-3,1\right),\left(1,0\right)\}$
  
  • (coefficient, exponent) 对；

1. 加法和乘法运算

p=p0+p1X+p2X2+⋯+pm−1Xm−1+pmXmq=q0+q1X+q2X2+⋯+qn−1Xn−1+qnXn$$\begin{array}{rl}& p=p_0+p_{1}X+p_2{X}^2+\cdots +p_{m-1}{X}^{m-1}+p_m{X}^m\\ & q=q_0+q_{1}X+q_2{X}^2+\cdots +q_{n-1}{X}^{n-1}+q_n{X}^n\end{array}$$

• 加法：p+q=r0+r1X+r2X2+⋯+rkXk$p+q=r_0+r_{1}X+r_2{X}^2+\cdots +r_k{X}^k$，
  
  • ri=pi+qi$r_i=p_i+q_i$
• 乘法：pq=s0+s1X+s2X2+⋯+slXl$pq=s_0+s_{1}X+s_2{X}^2+\cdots +s_l{X}^l$
  
  • si=p0qi+p1qi−1+…+piq0=∑kpkqi−k$s_i=p_0{q}_i+p_1{q}_{i-1}+\dots +p_i{q}_0=\sum _k{p}_k{q}_{i-k}$，如 weixin_42849849 所言，此为 {pi},{qi}$\{p_i\},\{q_i\}$ 两序列的卷积和；