# Polynomials in One Variable

## Polynomials

An expression of two or more than two algebraic terms that contain variable(s) that are raised to non-negative integral powers are called polynomials.

## Types of Polynomials

*Based on the number of terms a polynomial can be classified into monomial, binomial, trinomial, etc.*

- An algebraic expression having only
**one term**is called a**monomial**. P(x)=x is a monomial. - Polynomials having
**two terms**are called**binomials**. P(x)=x2+2x has two terms, x2 and 2x.So, it is a binomial. - Polynomials having
**three terms**are called**trinomials**. P(x)=x4+3×2−4 has three terms, x4, 3×2 and −4.So, it is a trinomial. - An algebraic expression of the form P(x)=c, where
*c*is a constant is called**constant polynomial.** - The constant polynomial
**0**is called the**zero polynomial**.

## Degree of a Polynomial

The** degree **of a **polynomial **is the** highest degree **of its individual terms with non-zero coefficients. The** degree **of a **term **is the **sum of the exponents **of the variables that appear in it.

For a polynomial in **one variable**, the **highest power of the variable** in the polynomial is the **degree** of the polynomial. f(x)=x2−9×3+2×8−6 is a polynomial with degree 8 as the highest power to which *x *is raised is 8.

**Note:**

- The degree of a
**non-zero constant polynomial**is**zero**. - The degree of the
**zero polynomial**is**not defined**.

A **polynomial** *p(x)* in one variable *x* is an algebraic expression in *x* of the form *p(x)*=*a x _{n }^{n }*+

*a x*

_{n}_{−1 }

^{n}^{−1 }+

*a x*

_{n}_{−2 }

^{n}^{−2 }+……..+

*a x*

_{2 }

^{2 }+

*ax a*

_{1 }+

_{0}, where

*a a a*

^{0 1 2}, , ……

*a*are constants i.

^{n}**x is a variable **

a ,a,a ……a_{0 1 2 n} are respectively the **coefficients** of x^{i}

Each of *a x a x a x _{n }^{n}*,

_{n}_{−1 }

^{n}^{−1},

_{n}_{−2 }

^{n}^{−2},……..

*a x axa*

_{2 }

^{2},

_{1 },

_{0},with

*a*≠ 0,is called a

_{n }**term**of a polynomial.

The highest power of the variable in a polynomial is called the **degree** of the polynomial.

- A polynomial with one term is called a
**monomial**. - A polynomial with two terms is called a
**binomial**. - A polynomial with three terms is called a
**trinomial**.

**★ **A polynomial with degree zero is called a **constant polynomial**.

** For example : **1, -3. The degree of non-zero constant polynomial is zero

**★ ** A polynomial of degree one is called a** linear polynomial**. It is of the form *ax* + *b*.

** For example : **

*x*– 2, 4

*y*+ 89, 3

*x – z*.

**★ ** A polynomial of degree two is called a **quadratic polynomial**. It is of the form *ax ^{2 }+ bx + c*. where

*a, b,*

*c*are real numbers and

*a*≠ 0

** For example : **

*x*

^{2 }− +2 5

*etc.*

_{x } **★ ** A polynomial of degree three is called a **cubic polynomial** andhas the general form *ax ^{3} + bx^{2} *+

*cx +d*.

**For example :** *x*^{3 }+ 2*x*^{2 }− +2*x *5 etc.

**★ ** A **bi-quadratic polynomial** *p(x*) is a polynomial of degree four which can be reduced to quadratic polynomial in the variable *z = x*^{2} by substitution.

**★ ** The **zero polynomial** is a polynomial in which the coefficients of all the terms of the variable are zero. Degree of zero polynomial is not defined.

** ★ **The

**value of a polynomial**f(x) at

*x*=

*p*is obtained by substituting

*x*=

*p*in the given polynomial and is denoted by

*f*(

*p*).

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

**★ ** The number of real zeroes of *a* polynomial is less than or equal to the degree of polynomial.

## Classification of Polynomials according to their Degree

*Polynomials can be classified on the basis of their degree as follows:*

- A polynomial of degree
**one**is called a**linear polynomial**. P(x)=x−2 is a linear polynomial. - A polynomial of degree
**two**is called a**quadratic polynomial**. P(x)=x2−3x+4 is a quadratic polynomial. - A polynomial of degree
**three**a cubic**polynomial**. P(x)=x3+3x−2 is a cubic polynomial.

## Representing equations on a graph

All polynomials can be represented on the graph to understand the nature of the polynomial, its zeroes etc.

**For example**, Geometrically **zeros **of a polynomial are the points where its graph **cuts the x-axis**.

## Zeroes of a Polynomial

A **zero** of a polynomial P(x) is a number *c* such that P(c)=0.

The zero’s of the polynomial P(x)=x2−4 are 2 and (-2) since P(2)=(2)2−4=0 and P(−2)=(−2)2−4=0.

**Note:**

- A
**non-zero constant**polynomial has no**zero**. - Every
**real number**is a zero of the**zero polynomial**.

## Number of zeroes

In general, a polynomial of degree *n* has at most *n* zeros.

- A
**linear**polynomial has**one**zero. - A
**quadratic**polynomial has at most**two**zeros. - A
**cubic**polynomial has at most**three**zeros.

# Remainder Theorem

## Long Division method to divide two polynomials

*To divide one polynomial by another, follow the steps given below.*

**Step 1: **arrange the terms of the dividend and the divisor in the decreasing order of their degrees. Suppose we want to divide (−x3+3×2+5−3x) by (−x2+x−1), we will arrange the terms of (−x3+3×2+5−3x) in decreasing order of their degrees as shown below.

**Step 2: **To obtain the first term of the quotient, divide the highest degree term of the dividend by the highest degree term of the divisor. Then carry out the division process. In our case, we will divide (−x3)(the highest degree term of the dividend) by (−x2)(the highest degree term of the divisor) to get the first term of the quotient. The first term of the quotient = −x3−x2 = x.

**Division algorithm** ** **

If *p(x*) and *g(x*) are the two polynomials such that degree of *p(x)* ≥ degree of *g(x)* and g*(x) ≠* 0, then we can find polynomials *q(x*) and *r(x)* such that:

*p (x)* = *g(x) q(x) + r(x)* where, *r(x)* =0 or degree of *r(x)* < degree of *g(x)*

**Step 3: **The remainder of the previous division becomes the dividend for the next step. Repeat this process until the degree of the remainder is less than the degree of the divisor.

*Dividing one polynomial by another polynomial.*

**Remainder Theorem**

When a polynomial f(x) of degree **greater than or equal to one** is divided by a linear polynomial x−a the remainder is equal to the value of f(a).

If f(a)=0 then x−a is a factor of the polynomial f(x).

**Factor Theorem**

If P(x) is a polynomial of degree *greater than or equal to one* and *a* is any real number then x−a is a factor of P(x) if P(a)=0.

**Remainder theorem** ** **

Let *p*(*x*) be any polynomial of degree greater than or equal to one and let a be any real number. If *p*(*x*) is divided by the linear polynomial (*x – a*), then remainder is *p*(*a*).

If the polynomial *p(x*) is divided by (*x+a*), the remainder is given by the value of* p *(-*a*). ii. If p (x) is divided by *ax + b =* 0*; a *≠* 0*, the remainder is given by

**Factor theorem** ** **

Let *p*(*x*) is a polynomial of degree *n *≥ 1and *a* is any real number such that *p*(*a*) = 0, then (*x* – *a*) is a factor of *p*(*x*).

**Converse of factor theorem** ** **

Let *p*(*x*) is a polynomial of degree *n* ≥ 1and *a* is any real number. If (*x* – *a*) is a factor of *p*(*x*), then *p*(*a*) = 0.

(*x* + *a*) is a factor of a polynomial *p*(*x*) iff *p*(-*a*) = 0. ii. (*ax* – *b*) is a factor of a polynomial *p*(*x*) iff *p*(*b*/*a*) = 0. iii. (*ax* + *b*) is a factor of a polynomial *p*(*x*) iff *p*(-*b*/*a*) = 0.

(*x* – *a*)(*x* – *b*) is a factor of a polynomial *p*(*x*) iff *p*(*a*) = 0 and *p*(*b*) = 0.

For applying factor theorem the divisor should be either a linear polynomial of the form (*ax* + *b*) or it should be reducible to a linear polynomial.

If *f*(*x*) is a polynomial with integral coefficients and the leading coefficient is 1, then any integer root of *f*(*x*) is a factor of the constant term.

# Factorization of Polynomials

## Factorisation of Quadratic Polynomials- Splitting the middle term

*Factorisation of the polynomial ax2+bx+c by splitting the middle term is as follows:*

**Step 1: **We split the middle term by finding two numbers such that their sum is equal to the coefficient of x and their product is equal to the product of the constant term and the coefficient of x2.

**For example **for the quadratic polynomial (x2+5x+6) the middle term can be split as, x2+2x+3x+6

Here,2+3=5 and 2×3=6.

**Step 2: **Now, we factorise by pairing the terms and taking the common factors.

x2+2x+3x+6=x(x+2)+3(x+2)=(x+2)(x+3)

Thus, x+2 and x+3 are factors of x2+5x+6.

## Factorisation of Quadratic Polynomials – Factor theorem

To factorise a quadratic polynomial f(x)=ax2+bx+c, find two numbers p and q such that f(p)=f(q)=0. Let us factorise the quadratic polynomial f(x)=x2−3x+2.

**(i) **f(2)=22−3(2)+2=4−6+2=0

Hence, x−2 is a factor of x2−3x+2.

**(ii)** f(3)=32−3×3+2=9−9+2=2≠0

Hence, x−3 is not a factor of x2−3x+2.

**(iii)** f(1)=12−3×1+2=0

Hence, x−1 is a factor of x2−3x+2.

So, x−1 and x−2 are the factors of the quadratic polynomial x2−3x+2.

x2−3x+2=(x−2)(x−1)

A quadratic polynomial *ax ^{2} + bx+ c* is

**factorised by splitting the middle term**by writing

*b*as

*ps + qr*such that (

*ps)*(

*qr*)

*= ac*.

Then, *ax ^{2} + bx+ c* = (px + q) (rx + s)

An **algebraic identity** is an algebraic equation which is true for all values of the variables occurring in it.

*x*^{3 }+ + −*y*^{3 }*z*^{3 }3*xyz x y z x*= + +( )( ^{2 }+ + − − −*y*^{2 }*z xy yz zx*^{2 })

If *x y z*+ + = 0 then *x*^{3 }+ + =*y*^{3 }*z*^{3 }3*xyz* Here, *x, y* and *z* are variables.

## Algebraic Identities

- (a+b)
^{2}=a^{2}+2ab+b^{2} - (a-b)
^{2}=a^{2}-2ab+b^{2} - a
^{2}−b^{2}=(a−b)(a+b) - (x+a)(x+b)=x
^{2}+(a+b)x+ab - (a+b+c)
^{2}=a^{2}+b^{2}+c^{2}+2(ab+bc+ca) - (a+b)
^{3}=a^{3}+b^{3}+3ab(a+b) - (a−b)
^{3}=a^{3}−b^{3}−3ab(a−b) - a
^{3}+b^{3}+c^{3}−3abc=(a+b+c)(a^{2}+b^{2}+c^{2}−ab−bc−ca)

Some useful **quadratic identities**:

- (
*x*+ y)^{2 }=*x*^{2 }+ 2*xy*+ y^{2} - (
*x*− y)^{2 }=*x*^{2 }– 2*xy*+ y^{2} - (
*x y x y*− )( + = −)*x*^{2 }*y*^{2} - (
*x a x b x*+ )( + = + +)^{2 }+ (*ab x ab*) - (
*x + y + z*)^{2 }=*x*^{2}+*y*^{2}+*z*^{2}+^{ }2*xy*+ 2*yz*+ 2*zx* - Here
*x, y, z*are variables and*a, b*are constants.