Maths: How Factorisation Reveals Polynomial Roots

Polynomial factorization and root finding sit at the heart of Number and Algebra in Maths AA HL. When a cubic such as f(x) = x³ − 4x² + x + 6 is known to have a linear factor (x − 2), the Factor Theorem guarantees that dividing f(x) by that factor leaves no remainder, exposing a quadratic quotient of the form x² + bx + c. Finding b and c through polynomial long division, or by expanding and comparing coefficients, transforms an intimidating cubic into a product of simpler pieces. This matters because factorization and root finding are two views of the same relationship: each linear factor (x − r) corresponds to a root where f(r) = 0. Once the quadratic quotient is factorised further into two linear factors, the full structure of f(x) becomes visible, and solving f(x) = 0 reduces to reading off the values that make each factor zero. Mastering this chain — division, quotient, factorisation, roots — builds the foundation for higher-degree polynomials and rational functions later in the course.