What are the relationships between zeroes and coefficients of a quadratic polynomial $ax^2 + bx + c$?

If \alpha and \beta are the zeroes of p(x) = ax^2 + bx + c (a \neq 0), then the sum of zeroes is \alpha + \beta = -\frac{b}{a} = -\frac{\text{coefficient of } x}{\text{coefficient of } x^2}, and the product of zeroes is \alpha \beta = \frac{c}{a} = \frac{\text{constant term}}{\text{coefficient of } x^2}.

How do you form a quadratic polynomial when the sum $S$ and product $P$ of its zeroes are given?

A quadratic polynomial with sum of zeroes S = \alpha + \beta and product of zeroes P = \alpha \beta is given by k[x^2 - Sx + P], where k is any non-zero real constant.

How do you evaluate symmetric expressions like $\alpha^2 + \beta^2$ and $\frac{1}{\alpha} + \frac{1}{\beta}$ without finding $\alpha$ and $\beta$ separately?

Use algebraic identities: \alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta = S^2 - 2P, and \frac{1}{\alpha} + \frac{1}{\beta} = \frac{\alpha + \beta}{\alpha\beta} = \frac{S}{P}. Substituting S = -\frac{b}{a} and P = \frac{c}{a} gives the exact numerical value directly.

Why does a parabola $y = ax^2 + bx + c$ open upwards when $a > 0$ and downwards when $a < 0$?

As |x| becomes very large, the ax^2 term dominates the polynomial. If a > 0, y \to +\infty as x \to \pm\infty, resulting in an upward-opening curve with a minimum point. If a < 0, y \to -\infty, producing a downward-opening curve with a maximum vertex.

Is polynomial long division and the Division Algorithm tested in CBSE 2026–27?

No, the Division Algorithm for polynomials and long division problems have been removed from the current CBSE syllabus. The active focus is strictly on the geometrical meaning of zeroes and relationship between zeroes and coefficients of quadratic polynomials.

Can a quadratic polynomial have no real zeroes? What does its graph look like?

Yes, if the discriminant D = b^2 - 4ac 0) or entirely below the x-axis (if a < 0) without touching or intersecting it.

All Revision Notes

MathematicsCh-2 15 min comprehensive revision

NCERT Class 10 Mathematics — Chapter 2

Polynomials

Geometrical meaning of zeroes, parabolic curves, relations between zeroes and coefficients of quadratic polynomials, and symmetric algebraic identities.

Quick Key Takeaways:

Degree and Zeroes: A polynomial

p(x)

of degree

n

has at most

n

real zeroes, corresponding to the number of points where the graph

y = p(x)

intersects the X-axis.

Quadratic Zeroes Relations: For

p(x) = ax^2 + bx + c

with zeroes

\alpha, \beta

\alpha + \beta = -\frac{b}{a}, \quad \alpha\beta = \frac{c}{a}

Polynomial Formation: A quadratic polynomial with zeroes

\alpha, \beta

is given by

k[x^2 - (\alpha + \beta)x + \alpha\beta]

, where

k \neq 0

Symmetric Expressions:

\alpha^2 + \beta^2 = (\alpha+\beta)^2 - 2\alpha\beta

and

\frac{1}{\alpha} + \frac{1}{\beta} = \frac{\alpha+\beta}{\alpha\beta}

Graph & Coordinate VisualizerExplore the geometric meaning of zeroes as x-intercepts of polynomial curves

Open Tool

Jump to Concept:1. 1. Geometrical Zeroes, Parabolic Curves & Degree Theorems 2. 2. Relationship Between Zeroes and Coefficients & Symmetric Identities 3. 3. Important Solved Board Examination Questions (3-Mark & 5-Mark)4. 4. CBSE Case-Study Modeling: Highway Overpass Parabolic Arch Modeling 5. 5. CBSE Examiner Marking Scheme, Calculation Traps & Step Rubric

1. Geometrical Zeroes, Parabolic Curves & Degree Theorems

Axioms & Foundational Theory

Comprehensive mathematical foundations, definitions, formulas, and visual conceptual model for Polynomials.

• Geometrical Meaning of the Zeroes of a Polynomial

•

Definition of a Zero: A real number

k

is a zero of polynomial

p(x)

p(k) = 0

•

Graphical Intersection with X-Axis: The real zeroes of

y = p(x)

are precisely the X-coordinates of the points where the graph intersects or touches the X-axis.

•

Linear Polynomial $ax + b$ : Graph is a straight line intersecting the X-axis at exactly one point:

(-\frac{b}{a}, 0)

•

Quadratic Polynomial $ax^2 + bx + c$ ( $a \neq 0$ ): Graph is a parabola opening upwards if

a > 0

, or downwards if

a < 0

•

Case 1 ( $D > 0$ ): Intersects X-axis at two distinct points

\implies

2 distinct real zeroes.

•

Case 2 ( $D = 0$ ): Touches X-axis at exactly one point

\implies

2 equal real zeroes.

•

Case 3 ( $D < 0$ ): Completely above or below X-axis

\implies

No real zeroes.

📊 Polynomials: Parabolic Zeroes & Geometric RootsVisual Model

Visual schematic mapping the geometry of parabolas, X-axis intersection roots, and the algebraic zero-coefficient relations.

2. Relationship Between Zeroes and Coefficients & Symmetric Identities

Theorem Proofs & Derivations

Step-by-step rigorous geometric and algebraic theorem proofs and formula derivations for Polynomials.

• Derivation of Zeroes-Coefficient Relationship for Quadratic Polynomials

•

Let

\alpha

and

\beta

be the zeroes of

p(x) = ax^2 + bx + c

(

a \neq 0

•

By Factor Theorem,

(x - \alpha)

and

(x - \beta)

are factors of

p(x)

•

ax^2 + bx + c = k(x - \alpha)(x - \beta) = k[x^2 - (\alpha + \beta)x + \alpha\beta] = kx^2 - k(\alpha+\beta)x + k\alpha\beta

•

Equating coefficients of like powers of

x

on both sides:

•

x^2

a = k

•

x

b = -k(\alpha + \beta) = -a(\alpha + \beta) \implies \mathbf{\alpha + \beta = -\frac{b}{a}}

•

Constant term:

c = k\alpha\beta = a\alpha\beta \implies \mathbf{\alpha\beta = \frac{c}{a}}

• High-Yield Symmetric Polynomial Identities

•

\alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta = \left(-\frac{b}{a}\right)^2 - 2\left(\frac{c}{a}\right) = \frac{b^2 - 2ac}{a^2}

•

(\alpha - \beta)^2 = (\alpha + \beta)^2 - 4\alpha\beta = \frac{b^2 - 4ac}{a^2} \implies |\alpha - \beta| = \frac{\sqrt{b^2 - 4ac}}{|a|}

•

\alpha^3 + \beta^3 = (\alpha + \beta)^3 - 3\alpha\beta(\alpha + \beta) = -\frac{b^3}{a^3} + \frac{3bc}{a^2} = \frac{3abc - b^3}{a^3}

•

\frac{\alpha}{\beta} + \frac{\beta}{\alpha} = \frac{\alpha^2 + \beta^2}{\alpha\beta} = \frac{b^2 - 2ac}{ac}

3. Important Solved Board Examination Questions (3-Mark & 5-Mark)

Topper Step Solutions

Standard CBSE board exam numericals and riders with complete step-by-step mathematical working and justifications.

• 3-Mark Standard Board Question: If $\alpha$ and $\beta$ are the zeroes of the quadratic polynomial $f(x) = x^2 - 5x + k$ such that $\alpha - \beta = 1$ , find the value of $k$ .

•

Step 1 (Coefficients): In

f(x) = x^2 - 5x + k

a = 1, b = -5, c = k

•

Step 2 (Sum and Product):

\alpha + \beta = -\frac{-5}{1} = 5 \quad \text{--- (1)}

\alpha\beta = \frac{k}{1} = k \quad \text{--- (2)}

•

Step 3 (Using Identity): Given

\alpha - \beta = 1

(\alpha - \beta)^2 = (\alpha + \beta)^2 - 4\alpha\beta

(1)^2 = (5)^2 - 4(k)

1 = 25 - 4k \implies 4k = 24 \implies k = 6

•

Final Boxed Answer:

\mathbf{k = 6}

• 5-Mark Heavyweight Board Problem / Rider: Find the zeroes of the quadratic polynomial $p(x) = 6x^2 - 3 - 7x$ and verify the relationship between the zeroes and the coefficients.

•

Step 1 (Standard Form): Rewrite in descending powers of

x

p(x) = 6x^2 - 7x - 3

•

Step 2 (Factorization by Splitting Middle Term):
Product

= 6 \times (-3) = -18

; Sum

= -7

. Factors:

-9

and

+2

6x^2 - 9x + 2x - 3 = 3x(2x - 3) + 1(2x - 3) = (2x - 3)(3x + 1)

•

Step 3 (Zeroes):

2x - 3 = 0 \implies \alpha = \frac{3}{2}

3x + 1 = 0 \implies \beta = -\frac{1}{3}

•

Step 4 (Verification of Sum):
-

\text{LHS: } \alpha + \beta = \frac{3}{2} + \left(-\frac{1}{3}\right) = \frac{9 - 2}{6} = \frac{7}{6}

\text{RHS: } -\frac{b}{a} = -\frac{-7}{6} = \frac{7}{6} \implies \text{LHS} = \text{RHS}

•

Step 5 (Verification of Product):
-

\text{LHS: } \alpha\beta = \left(\frac{3}{2}\right)\left(-\frac{1}{3}\right) = -\frac{3}{6} = -\frac{1}{2}

\text{RHS: } \frac{c}{a} = \frac{-3}{6} = -\frac{1}{2} \implies \text{LHS} = \text{RHS}

•

Conclusion: Relationships are verified.

4. CBSE Case-Study Modeling: Highway Overpass Parabolic Arch Modeling

Case Study Mastery (4 Marks)

Real-world mathematical modeling scenario with structured multi-part questions and full step solutions.

• Practical Application Context: Highway Overpass Parabolic Arch Modeling

A highway overpass arch is constructed in the shape of a downward parabola given by the quadratic polynomial $h(x) = -x^2 + 2x + 8$ , where $h(x)$ represents the height of the arch in meters and $x$ represents the horizontal distance from the left support.

•

Q1: Find the horizontal span (width) between the arch foundations.

\rightarrow

Set

h(x) = 0 \implies -x^2 + 2x + 8 = 0 \implies x^2 - 2x - 8 = 0 \implies (x - 4)(x + 2) = 0 \implies x = 4, x = -2

. Horizontal distance span

= 4 - (-2) = \mathbf{6\text{ meters}}

•

Q2: Find the maximum height of the parabolic arch.

\rightarrow

Axis of symmetry

x = -\frac{b}{2a} = -\frac{2}{2(-1)} = 1

. Maximum height

h(1) = -(1)^2 + 2(1) + 8 = -1 + 2 + 8 = \mathbf{9\text{ meters}}

•

Q3: What are the sum and product of the zeroes of this polynomial?

\rightarrow

Sum

= -\frac{b}{a} = -\frac{2}{-1} = \mathbf{2}

. Product

= \frac{c}{a} = \frac{8}{-1} = \mathbf{-8}

5. CBSE Examiner Marking Scheme, Calculation Traps & Step Rubric

Important Solved Board Questions

Examiner step-marking allocations, common algebraic/arithmetic deduction traps, and presentation guidelines.

• Step-by-Step Marking Rubric & Presentation Guidelines

•

1 Mark: Reorder polynomial into standard form

ax^2 + bx + c

•

1 Mark: Correct splitting of middle term and factor extraction.

•

1 Mark: Calculation of zeroes

\alpha, \beta

•

1 Mark: Explicit verification steps for both sum (

\alpha+\beta = -b/a

) and product (

\alpha\beta = c/a

• Common Calculation Traps & Verification Checklist

•

Trap 1: Omitting the constant

k

when forming a polynomial

k[x^2 - (\alpha+\beta)x + \alpha\beta]

•

Trap 2: Forgetting the negative sign in

\alpha + \beta = -b/a

•

Trap 3: Misidentifying the number of zeroes from a graph by counting intersections with the Y-axis instead of the X-axis.

Authentic Board Question (3 Marks)Topic: Polynomials Algebraic Formulations & Exact Methods

Solve the algebraic equation, find unknown coefficients, and apply standard theorem methods for Polynomials.

Official CBSE Step-by-Step Marking Breakdown:

Step 1: Standard Form Equation & Algebraic Setup: Convert given conditions into standard algebraic form (

ax^2+bx+c=0

a_n = a+(n-1)d

, or system of equations).

1 Mark

Step 2: Step-by-Step Factorisation / Reduction Method: Apply formal algebraic method (splitting the middle term, quadratic formula, elimination, or AP summation).

1 Mark

Step 3: Final Solution & Domain Verification: State the final values of

x, n, d

, verifying against real-life boundary conditions (e.g.

n > 0

, positive speeds).

1 Mark

Model Student Answer (Target: Full 3/3 Marks):

To score full 3 marks on Polynomials in CBSE Mathematics:

1. Algebraic Setup: Express the given problem into standard algebraic form.
2. Solving Technique: Carry out step-by-step algebraic manipulation showing all factorization or formula substitution lines.
3. Boxed Result: State and box the final root values, discarding any non-viable negative or fractional answers where context dictates.

Examiner Mark Deduction Traps:

•Always check boundary conditions (e.g., number of terms

n

in an AP must be a positive natural number).

•Show all factorization steps clearly—do not jump directly from equation to roots.

High-Frequency Conceptual Doubts & FAQs

Curated answers to the most common questions asked by Class 10 students.

The zeroes of a polynomial

y = p(x)

correspond precisely to the

x

-coordinates of the points where the graph of the polynomial intersects the

x

-axis. A polynomial of degree

n

can intersect the

x

-axis at most

n

times, and therefore has at most

n

real zeroes.

Polynomials

1. Geometrical Zeroes, Parabolic Curves & Degree Theorems

2. Relationship Between Zeroes and Coefficients & Symmetric Identities

3. Important Solved Board Examination Questions (3-Mark & 5-Mark)

4. CBSE Case-Study Modeling: Highway Overpass Parabolic Arch Modeling

5. CBSE Examiner Marking Scheme, Calculation Traps & Step Rubric

Official CBSE Step-by-Step Marking Breakdown:

High-Frequency Conceptual Doubts & FAQs

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