Polynomials
The exam tests zeros of linear, quadratic and cubic polynomials, the division algorithm, and the relationships between coefficients and zeros, especially for quadratics.
Core rules
- If α is a zero of p(x), then p(α) = 0 and (x − α) is a factor of p(x).
- Division algorithm: p(x) = g(x)q(x) + r(x), where either r(x) = 0 or degree r(x) < degree g(x).
- For ax² + bx + c: α + β = −b/a and αβ = c/a.
- A quadratic polynomial with given zeros is k[x² − (sum of zeros)x + product of zeros], where k ≠ 0.
- For a cubic ax³ + bx² + cx + d: α + β + γ = −b/a, αβ + βγ + γα = c/a and αβγ = −d/a.
- The zeros of a quadratic are the x-intercepts of its graph.
Worked example
Find the zeros of 6x² − 7x − 3 and verify the coefficient relationships. Split the middle term: 6x² − 9x + 2x − 3 = 3x(2x − 3) + 1(2x − 3) = (2x − 3)(3x + 1). Zeros: x = 3/2 and x = −1/3. Sum = 3/2 − 1/3 = 7/6, which equals −b/a = −(−7)/6. Product = (3/2)(−1/3) = −1/2, which equals c/a = −3/6.
Traps
- Losing the minus sign in −b/a.
- Forgetting the constant k when reconstructing a polynomial from its zeros.
- Misstating the remainder condition in the division algorithm.
- Counting a repeated zero as two different zeros when stating the number of zeros.