IB Mathematics: Applications and Interpretation HL · IB AI HL · 25 min read
1. Roots of Polynomials★★☆☆☆⏱ 15 min
By the Fundamental Theorem of Algebra, a polynomial of degree $n$ with real coefficients has exactly $n$ roots when counting repeated roots and complex roots. Complex roots always come in conjugate pairs for polynomials with real coefficients, and you will mostly work with real roots in IB AI HL.
2. Remainder Theorem★★☆☆☆⏱ 15 min
The remainder theorem can be generalized for any linear divisor: when dividing by $(kx - a)$, the remainder is equal to $P\left(\frac{a}{k}\right)$.
Exam tip: If an exam question asks only for the remainder, always use the remainder theorem instead of long division to save 2-3 minutes per question.
3. Factor Theorem★★★☆☆⏱ 20 min
The factor theorem is the primary tool for factorizing higher-degree polynomials (up to degree 4) in IB AI HL. Once you find one root by testing, you can factor out the linear term and reduce the problem to solving a lower-degree polynomial, which is far simpler.
Common Pitfalls
Why: Signs are easily mixed up when extracting the root from a linear factor
Why: The Fundamental Theorem of Algebra counts repeated and complex roots, not just distinct real roots
Why: The theorem is often memorized only for the $(x-a)$ form, leading to errors for non-monic divisors
Why: Higher-degree polynomials have multiple roots, so finding one does not complete the problem