To Understand the concept of **Geometric Meaning of the Zeros of a Polynomial**. Let us consider linear polynomial ax + b. The graph of y = ax + b is a straight line.

For example : The graph of y = 3x + 4 is a straight line passing through (0, 4) and (2, 10).

(i) Let us consider the graph of y = 2x – 4 intersects the x-axis at x = 2. The zero 2x – 4 is 2. Thus, the zero of the polynomial 2x – 4 is the x-coordinate of the point where the graph y = 2x – 4 intersects the x-axis.

(ii) A general equation of a **linear polynomial** is ax + b. The graph of y = ax + b is a straight line which intersects the x-axis at . **Zero of the polynomial** ax + b is the x-coordinate of the point of intersection of the graph with x-axis.

(iii) Let us consider the quadratic polynomial x^{2} – 4x + 3. The graph of x^{2} – 4x + 3 intersects the x-axis at the point (1, 0) and (3, 0).

Zeroes of the polynomial x^{2} – 4x + 3 are the x-coordinates of the points of intersection of the graph with x-axis.

The shape of the graph of the quadratic polynomials is ∪ and the curve is known as parabola.

(iv) Now let us consider one more polynomial –x^{2} + 2x + 8. Graph of this polynomial intersects the x-axis at the points (4, 0), (–2, 0). **Zeroes of the polynomial** –x^{2} + 2x + 8 are the x-coordinates of the points at which the graph intersects the x-axis. The shape of the graph of the given **quadratic polynomial** is ∩ and the curve is known as **parabola**.

The zeroes of a quadratic polynomial ax^{2} + bx + c he x-coordinates of the points where the graph of y = ax^{2} + bx + c intersects the x-axis.

**Cubic polynomial :** Let us find out geometrically how many zeroes a cubic has. To understand this let consider cubic polynomial x^{3} – 6x^{2} + 11x – 6.

**Case 1 : **The graph of the cubic equation intersects the x-axis at three points (1, 0), (2, 0) and (3, 0). Zeroes of the given polynomial are the x-coordinates of the points of intersection with the x-axis.

**Case 2 : **The cubic equation x^{3} – x^{2} intersects the x-axis at the point (0, 0) and (1, 0). Zero of a polynomial x^{3} – x^{2} are the x-coordinates of the point where the graph cuts the x-axis. Zeroes of the cubic polynomial are 0 and 1.

**Case 3 : **Let us consider the equation of cubic polynomial y = x^{3}. Cubic polynomial has only one zero.

In short, we say that, a **cubic equation** can have 1 or 2 or 3 zeroes or any polynomial of degree three can have at most three zeroes.

**Important Note**: In general, **polynomial of degree** n, the graph of y = p(x) passes x-axis at most at n points. Therefore, a polynomial p(x) of degree n has at most n zeroes.

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