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Geometrical Meaning of Zeroes of a Polynomial

🎓 Class 10 Mathematics CBSE Theory Ch 2 — Polynomials ⏱ ~30 min
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This MCQ module is based on: Geometrical Meaning of Zeroes of a Polynomial

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Targeting Class 10 level in Algebra, with Intermediate difficulty.

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2.1 Introduction

In Class IX, you studied polynomials? in one variable and their degrees. Recall that if \(p(x)\) is a polynomial in \(x\), the highest power of \(x\) in \(p(x)\) is called the degree of the polynomial \(p(x)\).

For instance, \(4x + 2\) is a polynomial in the variable \(x\) of degree 1, \(2y^2 - 3y + 4\) is a polynomial in \(y\) of degree 2, \(5x^3 - 4x^2 + x - \sqrt{2}\) is a polynomial in \(x\) of degree 3, and \(7u^6 - \frac{3}{2}u^4 + 4u^2 + u - 8\) is a polynomial in \(u\) of degree 6.

Expressions like \(\dfrac{1}{x-1}\), \(\sqrt{x} + 2\), \(\dfrac{1}{x^2+2x+3}\), etc., are not polynomials.

Types by Degree
Degree 1 → Linear polynomial: e.g., \(2x - 3\), \(\sqrt{3}x + 5\), \(y + \sqrt{2}\).
Degree 2 → Quadratic polynomial: e.g., \(2x^2 + 3x - \frac{2}{5}\), \(y^2 - 2\). General form: \(ax^2 + bx + c\), where \(a \ne 0\).
Degree 3 → Cubic polynomial: e.g., \(2 - x^3\), \(x^3 + x^2 + x - 1\). General form: \(ax^3 + bx^2 + cx + d\), where \(a \ne 0\).

Zeroes of a Polynomial

Consider the polynomial \(p(x) = x^2 - 3x - 4\). Putting \(x = 2\), we get \(p(2) = 4 - 6 - 4 = -6\). The value \(-6\) is obtained by replacing \(x\) by 2 in \(p(x)\). Similarly, \(p(0) = -4\).

If \(p(x)\) is a polynomial in \(x\) and \(k\) is any real number, then the value obtained by replacing \(x\) by \(k\) in \(p(x)\) is called the value of \(p(x)\) at \(x = k\), denoted \(p(k)\).

Definition
A real number \(k\) is called a zero? of the polynomial \(p(x)\) if \(p(k) = 0\).

For example, \(p(-1) = 1 + 3 - 4 = 0\) and \(p(4) = 16 - 12 - 4 = 0\). So \(-1\) and \(4\) are zeroes of \(x^2 - 3x - 4\).

For a linear polynomial \(ax + b\), the zero is \(k = -\dfrac{b}{a}\). Thus the zero of a linear polynomial is related to its coefficients. Does this happen for other polynomials too? We will explore this question in this chapter.

2.2 Geometrical Meaning of the Zeroes of a Polynomial

You know that a real number \(k\) is a zero of the polynomial \(p(x)\) if \(p(k) = 0\). But why are the zeroes of a polynomial so important? To answer this, let us first look at the geometrical representations of linear and quadratic polynomials and the geometrical meaning of their zeroes.

Graph of a Linear Polynomial

Consider a linear polynomial \(ax + b\), \(a \ne 0\). The graph of \(y = ax + b\) is a straight line. For example, \(y = 2x + 3\) passes through the points \((-2, -1)\) and \((2, 7)\).

X Y O -2 -1 1 2 1 2 3 -1 (-2,-1) (2,7) (0,3) (-3/2, 0)
Fig. 2.1 — Graph of \(y = 2x + 3\). The line crosses the x-axis at \(\left(-\frac{3}{2},\,0\right)\), which is the zero of \(2x+3\).

From Fig. 2.1, the graph of \(y = 2x + 3\) intersects the x-axis mid-way between \(x = -1\) and \(x = -2\), at the point \(\left(-\frac{3}{2},\, 0\right)\). The zero of \(2x + 3\) is \(-\frac{3}{2}\). Thus, the zero of a linear polynomial is the x-coordinate of the point where the graph of \(y = ax + b\) intersects the x-axis.

Key Observation
For a linear polynomial \(ax + b\), \(a \ne 0\), the graph of \(y = ax + b\) is a straight line which intersects the x-axis at exactly one point, namely \(\left(-\frac{b}{a},\, 0\right)\). Therefore, a linear polynomial has exactly one zero.

Graph of a Quadratic Polynomial

Now let us look at the geometrical meaning of a zero of a quadratic polynomial. Consider \(p(x) = x^2 - 3x - 4\). Let us list a few values of \(y = x^2 - 3x - 4\):

\(x\)-2-1012345
\(y = x^2-3x-4\)60-4-6-6-406
X Y O -2 -1 1 2 3 4 5 1 2 6 -1 -3 (-1, 0) (4, 0) (-2, 6) (5, 6) (0,-4) (3,-4)
Fig. 2.2 — Graph of \(y = x^2 - 3x - 4\). The parabola? crosses the x-axis at \((-1, 0)\) and \((4, 0)\), giving zeroes \(-1\) and \(4\).

The zeroes of the quadratic polynomial \(x^2 - 3x - 4\) are the x-coordinates of the points where the graph of \(y = x^2 - 3x - 4\) intersects the x-axis. These curves are called parabolas. The shape opens upwards if \(a > 0\) and downwards if \(a < 0\).

Three Cases for the Graph of a Quadratic

Case (i): Two distinct zeroes (Fig. 2.3)

The graph cuts the x-axis at two distinct points A and A'. The x-coordinates of A and A' are the two zeroes of the quadratic polynomial.

X Y O A A' (i) X Y O A A' (ii)
Fig. 2.3 — Case (i): Parabola cuts x-axis at two distinct points (two zeroes).

Case (ii): One zero (repeated) (Fig. 2.4)

The graph cuts the x-axis at exactly one point, i.e., at two coincident points. So the two points A and A' of Case (i) coincide to become one point A.

X Y O A (i) X Y O A (ii)
Fig. 2.4 — Case (ii): Parabola touches the x-axis at exactly one point (one repeated zero).

Case (iii): No real zeroes (Fig. 2.5)

The graph is either completely above the x-axis or completely below it. It does not cut the x-axis at any point.

X Y O (i) X Y O (ii)
Fig. 2.5 — Case (iii): Parabola does not intersect the x-axis (no real zeroes).
Important
A quadratic polynomial can have either two distinct zeroes, two equal (repeated) zeroes, or no real zeroes. In other words, a polynomial of degree? 2 has at most 2 zeroes.

Graph of a Cubic Polynomial

Now consider the cubic polynomial \(x^3 - 4x\). Let us find a few values:

\(x\)-2-1012
\(y = x^3-4x\)030-30
X Y O (-2, 0) (0, 0) (2, 0) (-1, 3) (1,-3)
Fig. 2.6 — Graph of \(y = x^3 - 4x\). The S-shaped curve crosses the x-axis at \(-2\), \(0\), and \(2\) — three zeroes.

We see that \(-2\), \(0\), and \(2\) are zeroes of the cubic polynomial \(x^3 - 4x\). They are the x-coordinates of the only points where the graph intersects the x-axis.

More Cubic Examples

Consider the cubic polynomials \(x^3\) and \(x^3 - x^2\). Their graphs are shown in Fig. 2.7 and Fig. 2.8.

X Y O (1, 1) (2, 8) (-1,-1) (-2,-8) Fig. 2.7 — \(y = x^3\) X Y O (-1,-2) (2, 4) Fig. 2.8 — \(y = x^3 - x^2\)
Fig. 2.7: \(y = x^3\) has one zero at origin. Fig. 2.8: \(y = x^3 - x^2\) has zeroes at 0 and 1.

Note that 0 is the only zero of the polynomial \(x^3\). Also, since \(x^3 - x^2 = x^2(x-1)\), the zeroes 0 and 1 are the only zeroes of \(x^3 - x^2\).

Remark
In general, given a polynomial \(p(x)\) of degree \(n\), the graph of \(y = p(x)\) intersects the x-axis at at most \(n\) points. Therefore, a polynomial of degree \(n\) has at most \(n\) zeroes.

Example 1

Look at the graphs in Fig. 2.9 given below. Each is the graph of \(y = p(x)\), where \(p(x)\) is a polynomial. For each of the graphs, find the number of zeroes of \(p(x)\).

XYO (i) XYO (ii) XYO (iii) XYO (iv) XYO (v) XYO (vi)
Fig. 2.9 — Six polynomial graphs for Example 1.
Solution for Example 1:
(i) The number of zeroes is 0 as the graph does not intersect the x-axis.
(ii) The number of zeroes is 2 as the graph intersects the x-axis at two points.
(iii) The number of zeroes is 3 as the graph intersects the x-axis at three points.
(iv) The number of zeroes is 1 as the graph intersects the x-axis at one point.
(v) The number of zeroes is 1 as the graph touches the x-axis at one point.
(vi) The number of zeroes is 4 as the graph intersects the x-axis at four points.

Exercise 2.1

Q1. The graphs of \(y = p(x)\) are given in Fig. 2.10 below, for some polynomials \(p(x)\). Find the number of zeroes of \(p(x)\), in each case.

XYO (i) XYO (ii) XYO (iii) XYO (iv) XYO (v) XYO (vi)
Fig. 2.10 — Six polynomial graphs for Exercise 2.1, Q1.
Answer for Exercise 2.1 Q1:
(i) The number of zeroes is 1 (the graph crosses the x-axis at one point).
(ii) The number of zeroes is 0 (the graph does not intersect the x-axis).
(iii) The number of zeroes is 3 (the graph crosses the x-axis at three points).
(iv) The number of zeroes is 2 (the graph crosses the x-axis at two points).
(v) The number of zeroes is 4 (the graph crosses the x-axis at four points).
(vi) The number of zeroes is 0 (the horizontal line does not intersect the x-axis).
Interactive: Parabola Explorer
Adjust \(a\), \(b\), and \(c\) in \(y = ax^2 + bx + c\) and see the graph and zeroes update in real time.
X Y -5 -3 -1 1 3 5
Equation: \(y = x^2 - 3x - 4\) | Zeroes: \(x = -1,\; x = 4\)
Activity: Visualising Zeroes from Graphs
L3 Apply
Materials: Graph paper, pencil, ruler
Predict: For the polynomial \(p(x) = x^2 - 4\), how many times will the graph cross the x-axis?
  1. Create a table of values for \(y = x^2 - 4\) using \(x = -3, -2, -1, 0, 1, 2, 3\).
  2. Plot the points on graph paper and draw a smooth curve through them.
  3. Mark the points where the curve crosses the x-axis. These x-coordinates are the zeroes.
  4. Verify by solving \(x^2 - 4 = 0 \Rightarrow x = \pm 2\). Do your graph results match?

Observation: The parabola \(y = x^2 - 4\) opens upward (since \(a = 1 > 0\)) and crosses the x-axis at exactly two points: \((-2, 0)\) and \((2, 0)\). This confirms that \(x^2 - 4\) has two zeroes: \(-2\) and \(2\). The vertex (lowest point) of the parabola is at \((0, -4)\).

Competency-Based Questions

Scenario: A ball is thrown upward from a building rooftop. Its height above the ground (in metres) after \(t\) seconds is modelled by \(h(t) = -5t^2 + 20t + 25\). The graph of \(h(t)\) is a downward-opening parabola.
Q1. At what time does the ball hit the ground?
L3 Apply
  • (a) \(t = 1\) s
  • (b) \(t = 4\) s
  • (c) \(t = 5\) s
  • (d) \(t = 25\) s
Answer: (c) \(t = 5\) s. The ball hits the ground when \(h(t)=0\). Solving \(-5t^2+20t+25=0\), i.e., \(t^2-4t-5=0\), gives \((t-5)(t+1)=0\), so \(t=5\) (rejecting \(t=-1\)).
Q2. What is the maximum height reached by the ball?
L3 Apply
Answer: 45 m. The vertex is at \(t = -\frac{b}{2a} = -\frac{20}{2(-5)} = 2\) s. \(h(2)=-5(4)+40+25=45\) m.
Q3. The polynomial \(p(x) = x^2 + 1\) has no real zeroes. What does this tell us about the shape of its graph?
L4 Analyse
Answer: Since \(x^2+1 \geq 1 > 0\) for all real \(x\), the parabola lies entirely above the x-axis. It never touches or crosses the x-axis. This corresponds to Case (iii) — no real zeroes.
Q4. A polynomial of degree 4 can have at most 4 zeroes. Sketch a possible graph of a degree-4 polynomial that has exactly 3 zeroes. Explain how this is possible.
L6 Create
Answer: A degree-4 polynomial can have exactly 3 zeroes if one of the zeroes is repeated. For example, \(p(x) = (x-1)^2(x+1)(x-3)\) has zeroes at \(x = -1, 1, 3\), but \(x = 1\) is a repeated zero. The graph touches (but does not cross) the x-axis at \(x=1\) and crosses at \(x=-1\) and \(x=3\).

Assertion-Reason Questions

Assertion (A): The polynomial \(p(x) = x^2 + 4\) has no real zeroes.
Reason (R): The graph of \(y = x^2 + 4\) does not intersect the x-axis.
(a) Both A and R are true, and R is the correct explanation of A.
(b) Both A and R are true, but R is not the correct explanation of A.
(c) A is true but R is false.
(d) A is false but R is true.
Answer: (a) — Since \(x^2 + 4 \geq 4 > 0\) for all real \(x\), the polynomial has no real zeroes. The graph stays above the x-axis. R correctly explains A because the zeroes of a polynomial are the x-coordinates where its graph meets the x-axis.
Assertion (A): A cubic polynomial always has at least one real zero.
Reason (R): The graph of a cubic polynomial must cross the x-axis at least once since it extends to \(+\infty\) in one direction and \(-\infty\) in the other.
(a) Both A and R are true, and R is the correct explanation of A.
(b) Both A and R are true, but R is not the correct explanation of A.
(c) A is true but R is false.
(d) A is false but R is true.
Answer: (a) — A cubic polynomial has odd degree. As \(x \to \infty\), \(y \to \pm\infty\) and as \(x \to -\infty\), \(y \to \mp\infty\). By the Intermediate Value Theorem, the graph must cross the x-axis at least once. R correctly explains A.
Assertion (A): The graph of \(y = x^2 - 6x + 9\) touches the x-axis at exactly one point.
Reason (R): \(x^2 - 6x + 9 = (x - 3)^2\), which has a repeated zero at \(x = 3\).
(a) Both A and R are true, and R is the correct explanation of A.
(b) Both A and R are true, but R is not the correct explanation of A.
(c) A is true but R is false.
(d) A is false but R is true.
Answer: (a) — \(x^2-6x+9 = (x-3)^2\) has a double root at \(x=3\). The parabola touches (but does not cross) the x-axis at the single point \((3,0)\). R correctly explains A.

Frequently Asked Questions — Polynomials

What is Geometrical Meaning of Zeroes of a Polynomial in NCERT Class 10 Mathematics?

Geometrical Meaning of Zeroes of a Polynomial is a key concept covered in NCERT Class 10 Mathematics, Chapter 2: Polynomials. This lesson builds the student's foundation in the chapter by explaining the core ideas with worked examples, definitions, and step-by-step methods aligned to the CBSE curriculum.

How do I solve problems on Geometrical Meaning of Zeroes of a Polynomial step by step?

To solve problems on Geometrical Meaning of Zeroes of a Polynomial, follow the NCERT method: identify the given quantities, choose the relevant formula or theorem, substitute values carefully, and simplify. Class 10 exercises gradually increase in difficulty — start with solved NCERT examples before attempting exercise questions, and always verify your answer by substitution or diagram.

What are the most important formulas for Chapter 2: Polynomials?

The essential formulas of Chapter 2 (Polynomials) are listed in the chapter summary and highlighted throughout the lesson in formula boxes. Memorise them and practise at least 2–3 problems per formula. CBSE board exams frequently test direct application as well as combined use of multiple formulas from this chapter.

Is Geometrical Meaning of Zeroes of a Polynomial important for the Class 10 board exam?

Geometrical Meaning of Zeroes of a Polynomial is part of the NCERT Class 10 Mathematics syllabus and appears in CBSE board exams. Questions typically include short-answer, long-answer, and competency-based items. Review the NCERT examples, exercise questions, and previous-year board problems on this topic to prepare confidently.

What mistakes should students avoid in Geometrical Meaning of Zeroes of a Polynomial?

Common mistakes in Geometrical Meaning of Zeroes of a Polynomial include skipping steps, misapplying formulas, sign errors, and losing track of units. Write each step clearly, double-check algebraic manipulations, and re-read the question after solving to verify that your answer matches what was asked.

Where can I find more NCERT practice questions on Geometrical Meaning of Zeroes of a Polynomial?

End-of-chapter NCERT exercises for Geometrical Meaning of Zeroes of a Polynomial cover all difficulty levels tested in CBSE exams. After completing them, try the examples again without looking at the solutions, attempt the NCERT Exemplar questions for Chapter 2, and solve at least one previous-year board paper to consolidate your understanding.

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