Polynomials — Quick Revision
The zeroes of a polynomial are the x-values where its graph meets the x-axis. A quadratic polynomial has at most 2 zeroes and a cubic has at most 3. For a quadratic ax² + bx + c, the sum of the zeroes is −b/a and their product is c/a. For a cubic ax³ + bx² + cx + d, the sum of the zeroes is −b/a, the sum of the products taken two at a time is c/a and the product of all three is −d/a. The division algorithm states that p(x) = g(x)·q(x) + r(x), where either r(x) = 0 or the degree of r(x) is less than the degree of g(x).
Key Points
- •Zeroes = points where the graph cuts the x-axis
- •Quadratic: at most 2 zeroes; Cubic: at most 3
- •Quadratic ax² + bx + c: sum = −b/a, product = c/a
- •Cubic ax³ + bx² + cx + d: sum = −b/a, product = −d/a
- •Division algorithm: p(x) = g(x)q(x) + r(x)