Polynomials: Zeros, Remainder and Factor Theorem
What the exam tests
Polynomials questions test degree, zeros, the remainder theorem, the factor theorem, and the relationship between the zeros and coefficients of a quadratic polynomial. You may also be asked to form a polynomial from given zeros.
Core rules
- A zero of p(x) is a number k such that p(k)=0.
- Remainder theorem: the remainder when p(x) is divided by x−k is p(k).
- Factor theorem: x−k is a factor of p(x) exactly when p(k)=0.
- For ax²+bx+c with zeros α and β: α+β=−b/a and αβ=c/a.
- A quadratic polynomial with given zeros is x²−(sum of zeros)x+(product of zeros), up to a non-zero constant factor.
- The degree is the highest power of x with a non-zero coefficient.
Worked example
Find the remainder when p(x)=2x³−3x+4 is divided by x−1. By the remainder theorem, the remainder is p(1)=2(1)³−3(1)+4=2−3+4=3. For q(x)=x²−5x+6, the zeros are 2 and 3 because q(x)=(x−2)(x−3). Check: sum=5=−(−5)/1 and product=6=6/1.
Traps
Use x−k, so dividing by x+2 means k=−2. Do not confuse a zero with the value of the polynomial at zero. In the sum of zeros, the sign is −b/a, not b/a. Missing terms have coefficient 0, so include them when comparing coefficients. A polynomial can have a constant multiplying all terms without changing its zeros.