The zeroes of a polynomial p(x) are the values of x for which p(x) = 0. Geometrically, zeroes are the x-coordinates of the points where the graph of y = p(x) intersects the x-axis.

For a quadratic polynomial ax² + bx + c with zeroes α and β: Sum of zeroes (α + β) = -b/a. Product of zeroes (αβ) = c/a. For a cubic polynomial: sum = -b/a, sum of products of pairs = c/a, product = -d/a.

The Division Algorithm for polynomials states: if p(x) and g(x) are polynomials with g(x) ≠ 0, then we can find q(x) (quotient) and r(x) (remainder) such that p(x) = g(x)·q(x) + r(x), where r(x) = 0 or degree of r(x) < degree of g(x). This is analogous to Euclid's division for integers. Application: if one zero of a cubic polynomial is known, divide the cubic by (x − zero) to get a quadratic, then solve it to find remaining zeros. Always verify: dividend = divisor × quotient + remainder.

The graph of y = p(x) gives a visual understanding of zeros. For a linear polynomial ax+b, the graph is a straight line cutting the x-axis at one point. For a quadratic ax²+bx+c, the parabola can cut the x-axis at 2 points (2 zeros), touch it at 1 point (1 repeated zero), or not cross it at all (0 real zeros). A cubic polynomial always cuts the x-axis at least once (1 or 3 real zeros). The shape of the parabola opens upward if a > 0 and downward if a < 0.