Students can go through AP 10th Class Maths Notes Chapter 2 Polynomials to understand and remember the concepts easily.

## Polynomials 10th Class Notes Maths Chapter 2

→ Polynomials are algebraic expressions that consist of variables and coefficients. Variables are also sometimes called indeterminates.

The word polynomial is derived from the Greek words ‘poly’ means ‘many’ and nominal’ means ‘terms’

→ Polynomial: A polynomial is defined as an expression which is composed of variables, constants and fexponents, that are combined using mathematical operations such as addition, subtraction, multiplication and division.

(Or)

Simply the algebraic sum of variable terms and constant terms is a polynomial.

Example :

i) Constants : 1, 2, 3,…

ii) Variables : g, h, x, y, -5x^{2},…

iii) Exponents : 5 in x^{5},…

→ Standard Form of a Polynomial :

P(x) = a_{n}x^{n} + a_{n – 1}x^{n – 1} + a_{n – 2}x^{n – 2} +….. + a_{1}x + a_{0}

Where a_{n}, a_{n – 1}, a_{n – 2}, …….. , a_{1} are called coefficients of x^{n}, x^{n – 1}, xn^{n – 2}, ……. x and a_{0} constant term respectively and it is a real number (a_{0} ∈ R).

→ Notation: The polynomial function is denoted by P(x) where x represents the variable.

For example : P(x) = 3x^{2} – 7x + 9

→ Types of Polynomials : Polynomials are classified into the following categories depending upon the number of terms they have.

→ Monomial : A monomial is an algebraic expression having only one term.

For an expression to be a monomial, the single term should be a non-zero term.

Examples : -6x, 9, 3a^{4}, -3xy

→ Binomial : A binomial is a polynomial which contains exactly two terms.

A binomial can be considered as a sum or difference between two or more monomials.

Example : -5x + 3, 6a^{4} + 17x, xy^{2} + xy

→ Trinomial : A trinomial is a polynomial having exactly three terms.

Example : -5a^{4} + 2a + 7, 7x^{2} + 11x – 7

→ Polynomial : In general a polynomial may contain two or more terms.

→ Degree of a monomial or a term: The degree of a monomial or a term is defined as the highest exponent of the variable within the term or the sum of exponents of all variables present with in the term.

Example : Degree of 3x is 1.

Degree of 3x^{2}y^{3} is 2 + 3 = 5

→ Degree of a Polynomial: The degree of a polynomial is defined as the highest exponent of a monomial with in that polynomial.

Degree of the polynomial 3x^{5} + 5x – 7 is 5.

→ Constant polynomial: A polynomial of degree zero is called a constant polynomial or a polynomial containing only constant terms is a zero polynomial.

Example : -9, 2/5, …….

→ Linear polynomial: A linear polynomial is a polynomial defined by an equation of the form p(x) = ax + b, where a and b are real numbers and a ≠ 0.

(Or)

A polynomial of degree 1 is called a linear polynomial.

Example : – 5x + 3, 7a – 5, 3x + 6, …….

→ Quadratic polynomial: A polynomial of degree 2 is called a quadratic polynomial.

A polynomial of the form ax^{2} + bx + c = 0 is called a quadratic polynomial, a ≠ 0.

Example : 5x^{2} – 8x + 4 = 0

→ Cubic polynomial: A polynomial of degree 3 is called a cubic polynomial.

(Or)

A polynomial of the form ax^{3} + bx^{2} + cx + d = 0 is called a cubic polynomial where a ≠ 0.

→ Bi-quadratic polynomial : A polynomial of degree 4 is called a bi-quadratic polynomial.

→ Value of a polynomial: If p(x) is a polynomial, then p(a) is called the value of p(x) at a where x is the variable of the polynomial and a is any real number.

Example : If p(x) = x^{2} + 5x + 2, then the value of p(3), p(2), p(0) and p(3) + p(2) + p(0).

Solution:

p(3) = 3^{2} + 5(3) + 2 = 26

p(2) = 2^{2} + 5(2) + 2 = 16

p(0) = 0^{2} + 5(0) + 2 = 2

⇒ p(3) + p(2) + p(0) = 26 + 16 + 2 = 44

→ Zero of a polynomial: The real value of the variable at which a polynomial becomes zero is called a zero of the polynomial.

If p(x) is a polynomial and p(a) = 0, then ‘a’ is a zero of the polynomial p(x).

Example : p(x) = 3x – 12

Now, take x = 4 in the polynomial, i.e., p(4) = 3(4) – 12 = 0.

Thus, x = 4 is a zero of polynomial p(x) = 3x – 12

Check whether x = 1 is zero of p(x) = x^{3} – 6x^{2} + 11x – 6

f(1) = (1)^{3} – 6(1)^{2} + 11(1) – 6

⇒ f(1) = 1 – 6 + 11 – 6 = 12 – 12 = 0

Thus, x = 1 is a zero of p(x).

→ Factor Theorem : For the polynomial P(x), the factor theorem states that if x = a is zero of P(x) if and only if x – a is a factor of P(x). i.e., both the following conditions should hold true.

If a is a zero of P(x), then x – a will be a factor of P(x).

If x – a is a factor of P(x), then a will be a zero of P(x).

→ Relation between Zeros and Coefficient: The relation between the zeros and the

coefficient of the quadratic and cubic equation is discussed below.

→ Relation between Zeros and Coefficient of a Quadratic Equation : α and β are the

two zeros of a quadratic equation of the form ax^{2} + bx + c = 0, then

i) Sum of roots = α + β = \(\frac{-b}{a}\) = \(\frac{\text {-coefficient of } x}{\text { coefficient of } x^2}\)

ii) Product of roots = α × β = \(\frac{c}{a}\) = \(\frac{\text { constant term }}{\text { coefficient of } \mathrm{x}^2}\)

→ Relation between Zeros and Coefficient of a Cubic polynomial: If α, β and γ are the

roots of the cubic polynomial ax^{3} + bx^{2} + cx + d = 0, then the relation between its zeros and coefficients is given as follows :

i) α + β + γ = \(\frac{-b}{a}\) = \(\frac{\text { Coefficient of } \dot{x}^2}{\text { Coefficient of } x^3}\)

ii) αβ + βγ + αγ = \(\frac{c}{a}\) = \(\frac{\text { Coefficient of } x}{\text { Coefficient of } x^3}\)

iii) α β γ = \(\frac{-d}{a}\) = \(\frac{- \text { Constant }}{\text { Coefficient of } x^3}\)

→ Forming Equation with Zeros of Polynomial: If α and β are the roots of a quadratic

polynomial, then it is given by x^{2} – (α + β)x + αβ.

If α, β and y are the roots of a cubic polynomial, then it is given by x^{3} – (α + β + γ)x^{2} + (αβ + βγ + αγ)x – αβγ

→ Graphs of a polynomial:

Geometrical Meaning of Zeros of a Polynomial: Geometrically, zeros of a polynomial are the points, where its graph cuts the X-axis,

i) One zero (Linear Polynomial)

ii) Two zeros (Quadratic Polynomial)

iii)Three zeros (Cubic Polynomial)

Here A, B and C represent the zeros of a Linear Polynomial, Quadratic Polynomial and a Cubic polynomial.

→ Number of Zeros : In general, a polynomial of degree n has at most n zeros.

1) A linear polynomial has one zero.

2) A quadratic polynomial has at most two zeros.

3) A cubic polynomial has at most 3 zeros.

→ Graph of a quadratic polynomial: Graph of a quadratic polynomial is a parabola. The shape of the parabola is determined by the coefficient a’ of the quadratic polynomial p(x) = ax^{2} + bx + c, where a, b, c are real numbers and a ≠ 0.

Steps:

1) The vertex form of a quadratic function is f(x) = a(x – h)^{2} + k, where (h, k) is the vertex of the parabola.

2) The coefficient a determines whether the graph of a quadratic function will open upwards or downwards.

→ Graphing Quadratic Functions in Vertex Form : Consider the general quadratic function p(x) = ax^{2} + bx + c.

First, we rearrange it (by the method of completion of squares to the following form : p(x) = a(x + b/2a)^{2} – D/4a.

The term D is the discriminant, given by D = b^{2} – 4ac.

Here, the vertex of the parabola is (h, k) = (-b/2a, -D/4a).

Now, to plot the graph of p(x), we start by taking the graph of x2 and applying a series of transformations to it :

Step 1 : x^{2} to ax^{2}

This is a process of vertical scaling of the original parabola.

If a is negative, the parabola will also flip its mouth from the positive to the negative side.

The magnitude of the scaling depends upon the magnitude of a.

Step 2 : ax^{2} to a(x + b/2a)^{2}

This is a horizontal shift of magnitude |b/2a| units.

The direction of the shift will be decided by the sign of b/2a.

The new vertex of the parabola will be at (-b/2a, 0). The following figure shows an example shift :

Step 3 :

a(x + b/2a)^{2} to a(x + b/2a) – D/4a :

This transformation is a vertical shift of magnitude |D/4a| units.

The direction of the shift will be decided by the sign of D/4a.

The final vertex of the parabola will be at (-b/2a, -D/4a).

The following figure shows an example shift :

The graph of quadratic functions can also be obtained using the graphing . qua-dratic functions calculator.

→ Graph of a cubic polynomial: A cubic function is a polynomial function of degree 3 and is of the form p(x) = ax^{3} + bx^{2} + cx + d, where a, b, c and d are real numbers and a ≠ 0.

So, the graph of a cube function may have a maximum of 3 roots, i.e., it may intersect the x-axis at a maximum of 3 points.

The basic cubic function is f(x) = x^{2}.

Since complex roots always occur in pairs, a cubic function always has either 1 or 3 real zeros.

It cannot have 2 real zeros.

A cubic function is of the form f(x) = ax^{3} + bx^{2} + cx + d.

Example :

→ Domain and Range of a Cubic Function: Since a cubic function y = f(x) is a polynomial function, it is defined for all real values of x and hence its domain is the set of all real numbers (R). Also all y-values are being covered by the graph and hence the range of a cubic function is the set of all numbers as well.

So, the domain of a cubic function is R, the range of a cubic function is R.

The x-intercepts of a function are also known as roots (or) zeros. As the degree of a cubic function is 3, it can have a maximum of 3 roots.

Since complex roots of any function always occur in pairs, a function will always have 0, 2, 4, … complex roots.

So a function can either have 0 or two complex roots.

Thus, it has one or three real roots or x-intercepts.

To find the x-intercept(s) of a cubic function, we just substitute y = 0 or p(x) = 0 and solve for x-values.