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

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