Zeroes of a Polynomial and Geometric Meaning for CBSE Class 10

One of the central questions in algebra is finding which input values cause an algebraic expression to evaluate to zero. In CBSE Class 10 Mathematics, this concept is formalized as the zeroes of a polynomial.

While finding zeroes algebraically is an essential skill, understanding their geometric meaning—how zeroes manifest visually as coordinate intersections on the Cartesian plane—connects algebra and geometry. This geometric viewpoint is heavily tested in CBSE board exams through graphical questions.

What You Will Learn

• Formal definition of a zero of a polynomial
• Distinction between the zero of a polynomial and the root of an equation
• Geometric meaning of zeroes: The $x$-intercepts of $y = P(x)$
• Graphical behavior of linear, quadratic (parabolas), and cubic polynomials
• The maximum number of real zeroes for a polynomial of degree $n$
• Step-by-step solutions to CBSE board graphical problems (NCERT Exercise 2.1)
• Key exam tips and common analytical traps

1. What is a Zero of a Polynomial?

Formal Definition

A real number $k$ is said to be a zero of a polynomial $P(x)$ if and only if: $P(k) = 0$

If substituting $x = k$ makes the value of the polynomial zero, then $k$ is a zero.

Example

Consider $P(x) = x^2 - 3x - 4$:

• Substitute $x = 4$: $P(4) = 4^2 - 3(4) - 4 = 16 - 12 - 4 = 0$. Therefore, $4$ is a zero of $P(x)$.
• Substitute $x = -1$: $P(-1) = (-1)^2 - 3(-1) - 4 = 1 + 3 - 4 = 0$. Therefore, $-1$ is also a zero of $P(x)$.

Important: <u>Zero of a polynomial is a value of the variable, NOT necessarily the number zero ($0$). A polynomial can have zeroes that are positive, negative, fractional, or zero itself. For example, the zero of $P(x) = 2x - 6$ is $x = 3$.</u>

2. Geometric Meaning of the Zeroes of a Polynomial

When we graph the equation $y = P(x)$ on the Cartesian coordinate plane:

• Every point on the graph has coordinates $(x, y) = (x, P(x))$.
• A zero of $P(x)$ is a value of $x$ where $P(x) = 0$, which means $y = 0$.
• Points where $y = 0$ lie precisely on the $x$-axis.

The Geometric Principle: <u>The real zeroes of a polynomial $P(x)$ are the $x$-coordinates of the points where the graph of $y = P(x)$ intersects or touches the $x$-axis.</u>

Consequently: $\text{Number of real zeroes of } P(x) = \text{Number of points where the graph intersects the } x\text{-axis}$

3. Graphical Behavior by Polynomial Degree

1. Linear Polynomial: $y = ax + b$ ($a \ne 0$)

• The graph of $y = ax + b$ is always a straight line.
• It intersects the $x$-axis at exactly one point: $\left(-\frac{b}{a}, 0\right)$.
• Therefore, every linear polynomial has exactly one zero, given by $x = -\frac{b}{a}$.

2. Quadratic Polynomial: $y = ax^2 + bx + c$ ($a \ne 0$)

The graph of a quadratic polynomial is a smooth U-shaped curve called a parabola.

• If $a > 0$, the parabola opens upwards ($\cup$).
• If $a < 0$, the parabola opens downwards ($\cap$).

Depending on the position of the parabola relative to the $x$-axis, three cases arise:

    Case 1: 2 Zeroes           Case 2: 1 Zero (Coincident)        Case 3: 0 Zeroes
      \       /                      \       /
       \_____/                        \_____/                          \_____/
---------+---+---------            ------+------                     -----------
        A   B                            A (touches)                (no intersection)

1. Case I (Intersects at Two Distinct Points): The parabola cuts the $x$-axis at two distinct points $A(x_1, 0)$ and $B(x_2, 0)$. $\implies$ The polynomial has two distinct real zeroes ($x_1$ and $x_2$).
2. Case II (Touches at Exactly One Point): The parabola touches the $x$-axis at a single point $A(x_1, 0)$. $\implies$ The polynomial has two equal (coincident) real zeroes (effectively one distinct real zero).
3. Case III (Does Not Meet the $x$-axis): The parabola lies completely above or completely below the $x$-axis. $\implies$ The polynomial has no real zeroes.

Remember: A quadratic polynomial can have at most 2 real zeroes (either 2, 1, or 0 real zeroes).

3. Cubic Polynomial: $y = ax^3 + bx^2 + cx + d$ ($a \ne 0$)

The graph of a cubic polynomial can cross the $x$-axis:

• At 3 distinct points $\implies 3$ real zeroes.
• At 2 points (touching at one, crossing at another) $\implies 2$ real zeroes.
• At 1 point $\implies 1$ real zero.
• It must cross the $x$-axis at least once!
• Therefore, a cubic polynomial has at most 3 real zeroes, and at least 1 real zero.

General Theorem

In general, a polynomial $P(x)$ of degree $n$ can have at most $n$ real zeroes.

4. Solved CBSE Board Questions (NCERT Exercise 2.1 Patterns)

Problem: The graphs of $y = P(x)$ are given below for some polynomials $P(x)$. Find the number of zeroes of $P(x)$ in each case:

1. Case A: The graph is a horizontal line parallel to the $x$-axis passing through $(0, 3)$.
   • Analysis: The line never intersects the $x$-axis.
   • Answer: $0$ zeroes (no real zeroes).
2. Case B: The graph intersects the $x$-axis at $(-2, 0)$, $(1, 0)$, and $(3, 0)$.
   • Analysis: The curve intersects the $x$-axis at 3 distinct points.
   • Answer: $3$ zeroes.
3. Case C: An upward-opening parabola touches the $x$-axis at $(4, 0)$.
   • Analysis: The graph touches the $x$-axis at exactly 1 point.
   • Answer: $1$ zero (two coincident zeroes).
4. Case D: The graph intersects the $y$-axis at $(0, -2)$ and does not cross the $x$-axis.
   • Analysis: Zeroes correspond strictly to intersections with the $x$-axis, not the $y$-axis.
   • Answer: $0$ zeroes.

5. Summary and Examination Tips

Exam Tip: In graphical questions, count ONLY the intersections with the $x$-axis. Points where the graph crosses the $y$-axis are irrelevant to finding the zeroes of $P(x)$!

Common Mistake: Writing that a quadratic polynomial always has 2 real zeroes. It has at most 2 real zeroes; if the parabola does not touch the $x$-axis, it has zero real zeroes.