Polynomials
Polynomials questions in HSLC General Mathematics check degree, zeros, the remainder and factor theorems, and the relationship between zeros and coefficients. Most marks are lost through sign slips rather than hard algebra.
Core rules
- The degree of a polynomial is the highest power of x with a non-zero coefficient.
- Remainder Theorem: when p(x) is divided by (x − a), the remainder is p(a). For divisor (x + a), evaluate p(−a).
- Factor Theorem: (x − a) is a factor of p(x) exactly when p(a) = 0.
- For ax² + bx + c with zeros α and β: α + β = −b/a and αβ = c/a.
- Division algorithm: p(x) = d(x)q(x) + r(x), where deg r < deg d.
Worked example
Find the remainder when p(x) = x³ − 3x² + 4x − 5 is divided by x − 2. By the Remainder Theorem, remainder = p(2) = 8 − 12 + 8 − 5 = −1. No long division is needed. If the question instead asked whether x − 2 is a factor, the non-zero remainder −1 proves it is not.
Traps
The divisor x + 3 corresponds to a = −3, not 3. A zero of a polynomial is a value of x, not the constant term. Do not assume every cubic has three real zeros: it may have one real and two complex zeros. When using α² + β² = (α + β)² − 2αβ, substitute the signs carefully because α + β already carries −b/a.