*An algebraic expression containing one or more terms is called a polynomial in which exponent of the variables must be a whole number.*

2*x*, *x*^{2}*−y *+ *yx *, 2*y*−5*yz *+*2, *are polynomials.

2*x*^{-2}*−y , *is not a polynomial.

5√*yz , *is not a polynomial.

## Polynomials in One Variable

Polynomials in one variable have only one variable.

2*x,*

*x*^{2} + *x*,

*y*^{3} −5*y *+ 4

#### Coefficient

The numerical factor (constant) of the term in a polynomial is called it’s coefficient.

In *x*^{3}* −*3*x*^{2} + 2*x* + 5*,*

the coefficient of *x*^{3} is 1,

the coefficient of *x*^{2} is *−*3,

*x*is 2

*,*

*x*

^{0}is 5.

### Constant Polynomials

Polynomial having only a constant term is called a constant polynomial.

3,−7, 9 etc.

#### Zero Polynomial

The constant polynomial 0 is called the zero polynomial.

### Denoting a Polynomial in one variable

If the variable in a polynomial is* **x*, we may denote the polynomial by *p(x**)*, or *q(x**)*, or *r(x**)*, etc.

*p(x)* = *x*^{3}* −*3*x*^{2} + 2*x *+ 5

*q(x)* = 3*x*^{2} – *x* + 9

*r(y) *= 5*y*^{6} – 4*y*^{2} – 6

*s(n)* = *n*^{2} + *n*

A polynomial can have any (finite) number of terms.

#### Monomials

Polynomials having only one term.

*p(x)* = *x*^{2}

*q(y)* = *y*^{3}

#### Binomials

Polynomials having two terms.

*p(x)* = *x*^{2}+ 2*x*

*q(y)* = *y*^{3}–6

#### Trinomials

Polynomials having three terms.

*p(x)* = *x*^{3}−3*x*^{2} + 2

*r(n)* = 3*n*^{2} – n + 9

### Degree of a Polynomial

The highest exponent of a variable in a polynomial (with one variable) is called degree of the polynomial.

Coefficient of the term containing highest power of the variable must be non-zero.

The degree of a polynomial 3*x*^{7} – 4*x*^{6} + x + 9 is 7,

The degree of a polynomial 5*y*^{6} – 4*y*^{2} – 6 is 6.

* The degree of a non-zero constant polynomial is zero. *

*The degree of the zero polynomial is not defined.*

#### Degree of a Polynomial with more than one variable

Power of variables are added in each term and then compared. Highest sum will be the degree of polynomial.

e.g., 5x^{2}y + 3x^{2}z^{2} + y^{3},

Sum of powers in 1st, 2nd & 3rd terms are (2+1), (2+2) & (0+3) respectively. Since, 4 is highest sum, degree of the polynomial is 4.

#### Linear Polynomial

A polynomial of degree one is called a linear polynomial.

*p(x)* = *x** −*2

*q(y)* = 2*y* + 5

* A linear polynomial in one variable will have at most 2 terms.*

##### General form of linear polynomial in *x *

a*x* + b,

Where a ≠ 0 and a and b are constants.

#### Quadratic Polynomial

A polynomial of degree two is called a quadratic polynomial.

*p(x)* = *x*^{2}* −3x *^{2} + 2

*r(n)* = 3*n*^{2} – n + 9

*A quadratic polynomial in one variable will have at most 3 terms. *

##### General form of quadratic polynomial in *x*

*ax*^{2} + b*x* + c,

Where a ≠ 0 and a, b and c are constants.

#### Cubic Polynomial

A polynomial of degree three is called a cubic polynomial.

*p(x)* = 3*x*^{3} – 4*x*^{2} + x + 9

*q(y)* = 5*y*^{3} – 2*y* – 6

*A cubic polynomial in one variable will have at most 4 terms. *

#### General form of cubic polynomial in *x*

a*x*^{3} + b*x*^{2} + c*x* + d,

Where, a ≠ 0 and a, b, c and d are constants.

### Evaluating Polynomials

To find the value of a polynomial at the given value of the variable, we put the value of the variable in the polynomial.

Let’s find the value of *p(x)* = 3*x*^{3} – 4*x*^{2} + *x* + 9 at *x* =2

To find the value of *p* (*x*) at *x* = 2 we put 2 in the place of x,

*p*(2) = 3⋅2^{3} – 4⋅2^{2} + 2 + 9

= 3⋅8 – 4⋅4 + 2 + 9

= 24 – 16 + 11

= 19

Let’s find the value of* p(x)* = *x* −2 at *x* =2

*p*(2) = 2* −*2

= 0

### Zeroes (Roots) of a Polynomial

Value of the variables for which the value of a polynomial is zero is called zero of the polynomial.

A real number ‘a’ is a zero of a polynomial *p(x)* if *p(a)* = 0.

A zero of the polynomial *p(x)* is also called the * root *of the *polynomial equation,* *p(x) * =0.

Finding the zero of a polynomial *p(x)*, is the same as solving the equation *p(x)* = 0

Let’s find the zero of* p(x)* = 2*x* −4

*p(x)* = 0

⇒ 2*x* −4 = 0

⇒ 2*x* = 4

⇒ *x* = 4/2

⇒ *x* = 2

∴ 2 is the zero of the polynomial *p(x)* and, root of the equation *p(x)*=0.

### Root of a linear polynomial in one variable in general form

a*x* + b = 0 (general form of a linear polynomial in one variable)

⇒ a*x *= − b

⇒ *x = − b/a*

*Every linear polynomial in one variable has a unique zero.*

*A non-zero constant polynomial has no zero.*

e.g., 5*x*^{0} = 5 for any value of *x.*

*Every real number is a zero of the zero polynomial.*

e.g., 0*x* = 0 for any value of *x.*

⏪Division & Factorisation of Algebraic Expressions |
Division & Factorisation of Polynomials⏩ |