General Polynomial Equations

1. When Two or More Linear Equations are Multiplied, they give rise to an Equation of a Higher Degree.
2. Multiplying 2 equations gives a Quadratic Equation (Polynomial Equation of Degree 2), 3 equations gives a Cubic Equation (Polynomial Equation of Degree 3), 4 equations gives a Quartic Equation (Polynomial Equation of Degree 4) and so on.
3. In general, Multiplying \(R\) Linear Equations of Same \(N\) Variables each gives rise to a \(R\) Degree Polynomial Equation having Maximum Number of Terms given as
   
   Max No. of Terms = \( \frac{(R+1) \times (R+2) \times (R+3)\times \cdots \times(R+N)}{N!} \\ =\frac{(N+R) \times (N+R-1) \times (N+R-2) \times \cdots \times (R+2) \times (R+1)}{N!} \\ =\frac{(N+R) \times (N+R-1) \times (N+R-2) \times \cdots \times (R+2) \times (R+1) \times R \times (R-1) \cdots \times 2 \times 1}{R!N!} \\ =\frac{(N+R)!}{R!N!}=C(N+R,R)=C(N+R,N)\)   ...(1)
4. Maximum Number of Terms of Degree \(K\) (where \(K \leq R \) ) in a Polynomial Equation is given by
   
   Max No. of Terms of Degree K = \(\frac{(K+1) \times (K+2) \times (K+3)\times \cdots \times (K+N-1) \times (K+N)}{N!} - \frac{K \times (K+1) \times (K+2) \times \cdots \times(K+N-1)}{N!} \\ = \frac{(K+1) \times (K+2) \times (K+3)\times \cdots \times (K+N-1) \times( (K+N) -K)}{N!} \\ = \frac{(K+1) \times (K+2) \times (K+3)\times \cdots \times (K+N-1) \times N}{N!}=\frac{(K+1) \times (K+2) \times (K+3)\times \cdots \times (K+N-1)}{(N-1)!}\\ =\frac{(N+K-1) \times (N+K-2) \times \cdots \times (K+2) \times (K+1)}{(N-1)!} \\ =\frac{(N+K-1) \times (N+K-2) \times \cdots \times (K+2) \times (K+1) \times K \times (K-1) \cdots \times 2 \times 1}{K!(N-1)!} \\ =\frac{(N+K-1)!}{K!(N-1)!}=C(N+K-1,K)=C(N+K-1,N-1)\)   ...(2)
5. For Homogeneous Equations (i.e. Equations having only Terms that have Same Power as Degree of Polynomial), \(K=R\) and hence
   
   No. of Terms of in a Homogeneous Equation of Degree R =\(\frac{(N+R-1)!}{R!(N-1)!}=C(N+R-1,R)=C(N+R-1,N-1)\)   ...(3)
   
   Since Homogeneous Equations of \(N\) Variables and Degree \(R\) are analogous to Multinomial Expressions of \(N\) Variables and Degree \(R\), the formula in equation (3) also gives the Number of Terms in Multinomial Expansion of \(N\) Variables and Degree \(R\).