What is the easiest way to identify a zero from a graph?

Look for the x-coordinates where the graph crosses or touches the x-axis. Those x-values are the zeros.

Why is the discriminant important in quadratics?

It tells you whether the parabola has two real zeros, one repeated real zero, or no real zero.

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Chapter 2 | Class 10 | CBSE / NCERT

Polynomials

Complete Class 10 revision notes for degree, zeros, graphs, coefficient relations, and the division algorithm.

Chapter Roadmap

01What is a Polynomial?02Degree of a Polynomial 03Types of Polynomials 04Value of a Polynomial 05Zero of a Polynomial 06Graph of a Linear Polynomial 07Graph of a Quadratic Polynomial 08Three Cases of the Parabola 09Graph of a Cubic Polynomial 10Geometrical Meaning of Zeros 11Zeros and Coefficients (Quadratic)12Zeros and Coefficients (Cubic)13Finding a Polynomial from Its Zeros 14Division Algorithm

What is a Polynomial?

An algebraic expression with whole-number powers of the variable

Not a Polynomial

$\frac{1}{x}=x^{-1}$
$\sqrt{x}=x^{1/2}$
$2^x$

A polynomial is built from terms like

5x^2

3x

, and constants such as 2.

The power of the variable must be a non-negative integer. So negative powers, fractional powers, and variables in the exponent are not allowed.

Key Formulas

f(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0

All powers of the variable must be non-negative integers.

Practice

Which are polynomials:

3x^2-2x+1

x+\frac{1}{x}

\sqrt{2}x-5

x^{1/3}+4

?

Answer: The first and third.

Degree of a Polynomial

The highest power gives the degree

Examples

$2x-3$ has degree 1
$3x^2+x-4$ has degree 2
$2x^3-5x^2+x$ has degree 3

The degree is the highest power of the variable with a non-zero coefficient.

A non-zero constant has degree 0, while the zero polynomial has no defined degree.

Practice

Find the degree of

5x^3-3x^2+11

7y^4-\frac{3}{2}y

, and

13

Types of Polynomials

Classification by degree and number of terms

Degree 0 is constant, degree 1 is linear, degree 2 is quadratic, and degree 3 is cubic.

By terms: monomial means 1 term, binomial means 2 terms, and trinomial means 3 terms.

Key Formulas

ax+b

Linear polynomial

ax^2+bx+c

Quadratic polynomial

ax^3+bx^2+cx+d

Cubic polynomial

Practice

Classify

x^3-4x

7

, and

4x^2-5x+3

Value of a Polynomial

Substitute the given value of the variable

To find

f(a)

, replace every

x

with

a

and simplify.

This is called evaluating the polynomial at a point.

Worked Example

Evaluate a cubic

For

f(x)=2x^3-13x^2+17x+12

, find

f(1)

and

f(-2)

Show solution

$f(1)=2-13+17+12=18$

$f(-2)=-16-52-34+12=-90$

Practice

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

, find

p(0)

and

p(2)

Zero of a Polynomial

A value that makes the polynomial equal to zero

A real number

\alpha

is a zero of

f(x)

f(\alpha)=0

A polynomial of degree

n

can have at most

n

zeros.

Key Formulas

f(\alpha)=0

If this happens, then \alpha is a zero of f(x).

Practice

Find the zero of

2x-6

.

Answer:

x=3

Graph of a Linear Polynomial

A straight line with one x-intercept

The graph of

f(x)=ax+b

is a straight line.

Its zero is where the line cuts the x-axis.

Key Formulas

\text{Zero}=-\frac{b}{a}

The x-intercept of a line y=ax+b.

Graph of y = 2x - 5

A linear polynomial is always a straight line, and its zero is the single x-intercept.

Worked Example

Linear graph

For

f(x)=2x-5

, find the zero.

Show solution

Set $2x-5=0$ .

So $x=\frac{5}{2}$ .

Practice

Find the zero of

3x-9

Graph of a Quadratic Polynomial

A parabola that opens up or down

The graph of

ax^2+bx+c

is a parabola.

a>0

, it opens upward. If

a<0

, it opens downward.

Key Formulas

x_{\text{vertex}}=-\frac{b}{2a}

x-coordinate of the turning point

D=b^2-4ac

The discriminant decides the number of real zeros

Graph of y = x² - 2x - 8

This parabola opens upward, has vertex (1,-9), and crosses the x-axis at -2 and 4.

Practice

For

-x^2+2x+3

, state the opening direction and find the zeros.

Three Cases of the Parabola

The discriminant tells the number of real zeros

Cases

$D>0$ : two distinct real zeros
$D=0$ : one repeated real zero
$D<0$ : no real zero

For a quadratic, the discriminant is

D=b^2-4ac

It tells whether the parabola cuts, touches, or misses the x-axis.

D > 0

Two distinct real zeros

D = 0

One repeated real zero

D < 0

No real zeros

Practice

Compute

D

for

x^2+4x+4

x^2+1

, and

x^2-5x+6

Graph of a Cubic Polynomial

An S-shaped graph with at least one real zero

A cubic polynomial may cut the x-axis 1, 2, or 3 times.

It always has at least one real zero.

Three Zeros

x³ - 4x crosses the x-axis three times

Double Zero + Simple Zero

x³ - 2x² touches at 0 and crosses at 2

One Zero

x³ has one real zero at the origin

Worked Example

Factor and read the graph

Find the zeros of

x^3-4x

Show solution

$x^3-4x=x(x^2-4)=x(x-2)(x+2)$

Zeros are $-2$ , 0, and 2.

Practice

Find the zeros of

x^3-3x^2+2x

Geometrical Meaning of Zeros

Zeros are x-intercepts of the graph

The zeros of

f(x)

are the x-coordinates of the points where the graph

y=f(x)

meets the x-axis.

So a graph gives a direct visual meaning to zeros.

Key Formulas

\text{Zeros of }f(x)=\text{x-coordinates where }y=f(x)\text{ meets the x-axis}

Zeros From the Graph

The x-coordinates where the curve meets the x-axis are the zeros of the polynomial.

Zeros and Coefficients of a Quadratic Polynomial

Useful relations for sum and product of zeros

ax^2+bx+c

has zeros

\alpha

and

\beta

, then we can write direct relations using coefficients.

Key Formulas

\alpha+\beta=-\frac{b}{a}

Sum of zeros

\alpha\beta=\frac{c}{a}

Product of zeros

Worked Example

Verify a relation

For

x^2+7x+12

, find the zeros and verify the relations.

Show solution

$x^2+7x+12=(x+4)(x+3)$ , so zeros are $-4$ and $-3$ .

Sum $=-7$ and product $=12$ , matching the formulas.

Zeros and Coefficients of a Cubic Polynomial

Three standard relations for three zeros

For

ax^3+bx^2+cx+d

with zeros

\alpha

\beta

, and

\gamma

, three symmetric relations are used.

Key Formulas

\alpha+\beta+\gamma=-\frac{b}{a}

Sum of three zeros

\alpha\beta+\beta\gamma+\gamma\alpha=\frac{c}{a}

Sum of pairwise products

\alpha\beta\gamma=-\frac{d}{a}

Product of all three zeros

Finding a Polynomial from Its Zeros

Build the polynomial in reverse

If the zeros are known, a polynomial can be constructed from them.

For a quadratic with zeros

\alpha

and

\beta

, use the sum and product form.

Key Formulas

f(x)=k\left(x^2-(\alpha+\beta)x+\alpha\beta\right)

Use the given zeros to build a quadratic polynomial.

Practice

Find a quadratic polynomial with zeros

\frac{1}{4}

and

-1

.

Answer: One such polynomial is

4x^2+3x-1

Division Algorithm

Divide one polynomial by another

For polynomials

f(x)

and

g(x)

with

g(x)\neq 0

, division gives a quotient and remainder.

The remainder must have smaller degree than the divisor.

Key Formulas

f(x)=g(x)q(x)+r(x)

Dividend = Divisor x Quotient + Remainder

\deg(r)<\deg(g)

Remainder must have smaller degree than divisor

Worked Example

Polynomial division

Divide

6x^3+11x^2-39x-65

x^2+x-5

Show solution

The quotient is $6x+5$ and the remainder is $-14x-40$ .

Practice

Divide

x^3-3x^2+5x-3

x^2-2

.

Answer: Quotient

=x-3

, remainder

=7x-9

Quick Summary

Concept	Key Idea
Polynomial	Only non-negative integral powers of the variable
Degree	Highest power of the variable
Zero	$f(\alpha)=0$
Linear	Straight line, one zero
Quadratic	Parabola, up to two real zeros
Cubic	At least one real zero, at most three
Quadratic relation	$\alpha+\beta=-\frac{b}{a}$ and $\alpha\beta=\frac{c}{a}$
Division algorithm	$f(x)=g(x)q(x)+r(x)$

Polynomials: revision sheet

Close the notes and explain each idea in your own words. Then check the formulas and attempt the examples before opening their solutions.

Recall checklist

Polynomials and Their Degree
Zeros of a Polynomial — What They Mean Geometrically
Graphs of Linear, Quadratic, and Cubic Polynomials
Relationship Between Zeros and Coefficients
Division Algorithm for Polynomials
Common Mistakes and Board Answer Template

For each topic: state the rule, explain when it applies, give an example and identify one mistake to avoid. Return to any topic you cannot explain without looking.

Formula recap (3)

Relationship Between Zeros and Coefficients

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

Quadratic polynomial ax² + bx + c with zeros α and β.

Relationship Between Zeros and Coefficients

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

Cubic polynomial ax³ + bx² + cx + d with zeros α, β, γ.

Division Algorithm for Polynomials

p(x) = g(x) \cdot q(x) + r(x), \quad \deg(r) < \deg(g) \text{ or } r(x) = 0

Division algorithm for polynomials — analogous to a = bq + r for integers.

Explain your method

Write the given information, choose a method, show the steps, then check units and whether the answer makes sense.

Find the zeros of x^2 - 5x + 6 and verify the relation between zeros and coefficients.

Check worked solution 1

Factorise: x^2 - 5x + 6 = (x - 2)(x - 3). Zeros are 2 and 3. Their sum is 5 = -b/a and product is 6 = c/a.

Divide x^3 - 3x^2 + 5x - 3 by x^2 - 2.

Check worked solution 2

Quotient = x - 3 and remainder = 7x - 9.

Turn mistakes into revision

Record the question, the step that went wrong and the corrected reason. Try it again tomorrow without the solution, then revisit it after a week. A correct guess still needs an explanation.

Practise this chapter