The Binomial Theorem

The expansion of the expression

where \(n \in \mathbb{N}\) is given by the real polynomials

Theorem: Polynomial Division

Let \(A(x)\) and \(B(x)\) be two real polynomials such that the degree of \(A\) is greater than or equal to the degree of \(B\).

If \(B(x)\) is nonzero, then there exist unique polynomials \(Q(x)\) and \(R(x)\) such that

where

• \(\deg(Q) = \deg(A) - \deg(B)\)
• \(\deg(R) \lt \deg(B)\) or \(R(x)\) is the zero polynomial.

We call \(A\) the dividend, \(B\) the divisor, \(Q\) the quotient and \(R\) the remainder.

Theorem: Factorization Theorem

Every real polynomial \(A(x) = \sum_{k=0}^n a_k x^k\) can be factorized into a product of real polynomials of degree \(\le 2\). More specifically,

where:

• \(r_i\) are the distinct real roots of \(A(x) = 0\);
• \(p_i\) is the multiplicity of \(r_i\);
• \(c_j, d_j \in \mathbb{R}\);
• \(q_j \in \mathbb{N}_0\).

Polynomial Remainder Theorem (Little Bézout's Theorem)

The remainder \(R\) upon the division of a real polynomial \(A(x)\) with a real polynomial \(B(x) = x - p\) is the value of \(A\) at \(p\).