Polynomials
What this domain tests
Polynomials tests zeroes, the relationship between zeroes and coefficients, the division algorithm, and the remainder and factor theorems. AP SSC often asks you to verify coefficients from given zeroes or to find a remainder without long division.
Core rules
- A polynomial of degree n has at most n real zeroes. A zero is a value of x that makes p(x) = 0.
- For ax² + bx + c, sum of zeroes = −b/a and product = c/a.
- For ax³ + bx² + cx + d, sum = −b/a; sum of products taken two at a time = c/a; product = −d/a.
- Division algorithm: p(x) = g(x)q(x) + r(x), where the remainder r(x) is either zero or has degree less than g(x).
- Remainder theorem: the remainder on dividing p(x) by x − a is p(a).
- Factor theorem: x − a is a factor of p(x) exactly when p(a) = 0.
Worked example
Find the remainder when p(x) = x³ − 3x² + 2x − 1 is divided by x − 2. By the remainder theorem, substitute x = 2: p(2) = 8 − 12 + 4 − 1 = −1. So the remainder is −1. No long division is needed, and p(2) ≠ 0, so x − 2 is not a factor.
Traps
- Losing the minus sign in −b/a for the sum of zeroes.
- Using x = −2 for division by x + 2; the divisor x + 2 is x − (−2), so substitute −2.
- Confusing zeroes with coefficients.
- Saying a cubic must have three real zeroes; it has at most three.
- Ignoring the degree condition on the remainder in the division algorithm.