Polynomials, equations and integer properties
Polynomial and integer questions test exact algebra: factor and remainder theorems, rational-root candidates, divisibility, the Euclidean algorithm, and base representations. Neat bookkeeping is as important as the main theorem.
Core rules
- Remainder theorem: dividing P(x) by x-a leaves remainder P(a). Factor theorem: x-a is a factor exactly when P(a)=0.
- Rational root theorem: for integer coefficients, any rational root p/q in lowest terms has p dividing the constant term and q dividing the leading coefficient.
- Division algorithm: a=bq+r with 0≤r<b. The Euclidean algorithm repeatedly replaces (a,b) by (b,r); the last nonzero remainder is gcd(a,b).
- In base b, digits satisfy 0≤d<b, and the value is Σ d_k b^k. Convert by repeated division, reading remainders upward.
- For divisibility, use factorisation and modular arithmetic rather than testing many numbers.
Worked example
Factor P(x)=x^3-2x^2-5x+6. Test small integers: P(1)=1-2-5+6=0, so x-1 is a factor. Synthetic or polynomial division gives P(x)=(x-1)(x^2-x-6). The quadratic factors as (x-3)(x+2). Hence P(x)=(x-1)(x-3)(x+2), with roots 1,3,-2. The rational root candidates were divisors of 6 divided by divisors of 1, and all three roots appear among them.
Traps
A zero remainder confirms a factor, but a nonzero value must not be forced to zero. Include both positive and negative divisors in rational-root candidates. In the Euclidean algorithm, the gcd is the last nonzero remainder, not the final zero. In base conversion, remainders must be less than the base, and the order of digits is easy to reverse.