Polynomials — Quick Revision
A polynomial in x is an expression of the form a₀ + a₁x + a₂x² + … where the powers of x are whole numbers. The highest power is the degree: degree 1 is linear, 2 is quadratic and 3 is cubic. A number k is a zero of p(x) if p(k) = 0. The remainder theorem says that when p(x) is divided by (x − a), the remainder is p(a). The factor theorem follows from it: (x − a) is a factor of p(x) exactly when p(a) = 0. These, together with the standard identities, let us factorise polynomials quickly.
Key Points
- •Degree = highest power of the variable
- •Linear (1), Quadratic (2), Cubic (3)
- •k is a zero of p(x) if p(k) = 0
- •Remainder theorem: p(x) ÷ (x − a) leaves p(a)
- •Factor theorem: (x − a) is a factor ⇔ p(a) = 0