Mathematics | Class 10 Mathematics Notes

2.2 Geometrical Meaning of the Zeroes of a Polynomial

This section explains the importance of zeros of polynomials through their geometric interpretation as points where the graph of the polynomial intersects the x-axis. For a linear polynomial y = ax + b (a ≠ 0), the graph is a straight line intersecting the x-axis at one point (-b/a, 0), which corresponds to the zero of the polynomial. For example, y = 2x + 3 intersects the x-axis at (-3/2, 0). This is illustrated in Fig. 2.1 and Table 2.1, showing values of y for x = -2 and 2. For quadratic polynomials y = ax² + bx + c (a ≠ 0), the graph is a parabola opening upwards if a > 0 or downwards if a < 0. The zeros of the quadratic polynomial are the x-coordinates where the parabola intersects the x-axis. Using the example y = x² - 3x - 4, Table 2.2 lists y-values for various x, showing zeros at x = -1 and x = 4, confirmed by the graph in Fig. 2.2. Three cases arise for quadratic polynomials: (i) two distinct zeros where the parabola cuts the x-axis at two points (Fig. 2.3), (ii) one zero where the parabola is tangent to the x-axis (Fig. 2.4), and (iii) no zeros where the parabola lies entirely above or below the x-axis (Fig. 2.5). For cubic polynomials y = ax³ + bx² + cx + d, the graph can intersect the x-axis at up to three points, corresponding to up to three zeros. The example y = x³ - 4x has zeros at -2, 0, and 2, as shown in Table 2.3 and Fig. 2.6. Other examples include y = x³ and y = x³ - x² with zeros at 0 and 0,1 respectively (Figs. 2.7 and 2.8). The section concludes with the general remark that a polynomial of degree n has at most n zeros, corresponding to at most n x-axis intersections.

📊 Diagram: [figure_1] [figure_2] [figure_3] [figure_4] [figure_5] [figure_6] [figure_9] [figure_10] [table_1] [table_2] [table_3] Fig. 2.1; Fig. 2.2; Fig. 2.3; Fig. 2.4; Fig. 2.5; Fig. 2.6; Fig. 2.7; Fig. 2.8; Table on page 3 (2×3); Table on page 4 (2×9); Table on page 6 (2×6)

🧪 Activity: Students are encouraged to plot values of polynomials for various x on graph paper to visualize zeros as x-axis intersections.

🔗 Connection: Prepares for Exercise 2.1 on identifying zeros from graphs and leads to study of relationships between zeros and coefficients.