Basic concept of polynomial

Polynomial definition

A combination of constants and variables, connected by ‘ + , – , x & ÷ (addition, subtraction, multiplication and division) is known as an algebraic expression.

An algebraic expression in which the variables involves have only non-negative integral powers, is called polynomial or poly means many and nomial means terms .

Examples for polynomials:

x
20
x – 5
a2 + ab2 +25
x3 + 2x2 +10
3x2 + 3xy + 4y2 + 15
3xyz2 – 3x + 10z + 0.5

How to find the polynomial

Ex – 1: 20

Here “20” is just a constant and also having only one term, so it can be called as polynomial

Ex – 2: $\sqrt{5}$ is also be a polynomial because it is a constant (= 2.2360…etc)

Ex – 3: $\sqrt{y}$ is not a polynomial because the exponent of variable is “½”

Note: Exponents of variables in a polynomial allowed only 0, 1, 2, 3, … etc

Ex – 4: 4a-5 is not a polynomial because the exponent is “-5”

Ex – 5: $\frac{5}{a +2}$ is not a polynomial

Note: A polynomial never division by a variable

Ex-6 : $\frac{5a+ 2b + a^2}{8}$ is a polynomial

Note: A polynomial can divide by a constant but never division by a variable

Degree of polynomial:

In a polynomial the largest exponent value of any given variable, that value is degree of that polynomial.

Degree of a term is the sum of the exponents of its variable factors and degree of polynomial is the largest degree of its variable term.

Ex – 1 : 3x3 + 3z2 – 10z + 0.5

The terms of above polynomial are 3x3, 3z2 , 10z , 0.5

The coefficient of 3x3 is 3

The coefficient of 3z2 is 2

The coefficient of -10z is 1

The Degree of the above polynomial is 3

Ex – 2 : 8

In the above example contains constant number 8 and it can be written as 8x0

The degree of polynomialis zero

Note:

By adding or multiplying polynomials you get also a polynomial.
While writing a polynomial in a standard form, put the terms with the highest degree first.

Types of polynomials according to number of terms (Algebraic expressions)

Monomial

An algebraic expression containing only one term is called a monomial

Ex ; 7, 8x , $\frac{5 a^2 + b}{4}$ , x5

Here all algebraic expression containing one term

Binomial:

An algebraic expression containing two term is called a binomial

Ex ; 7 +x , 8x2 +y, 8a2 + 2ab , 25a – b2 , . . . . etc

Here all algebraic expression containing two terms

Trinomial:

An algebraic expression containing three term is called a trinomial

Ex ; y2 +x -7 , 8x2+y2+2xy, 8a2 + 2ab+25 , 25a2 – b2+ab , . . . . etc

Here all algebraic expression containing three terms

Multinomials:

An algebraic expression containing more than three term is called a multinomials

Ex ; y2+xy -x – 10 , 8x2+y2+2xy+x2y+12, 8a2 + 2ab+25 , 25a – b2+ab , . . . . etc

Here all algebraic expression containing more than three terms

Types of polynomials according to degree

Constant Polynomials

A polynomials having one term consisting of a constant only is called a constant polynomials

Ex ; 8 , $\sqrt{2}$ , 35 , . . . etc

Linear Polynomial

In a polynomial the largest exponent value of any variable is one then it is called liner polynomial

Ex: a + b, x + 25 , y + x + 25 . . . . etc

Quadratic Polynomial

In a polynomial the largest exponent value of any given variable is two then it is called Quadratic polynomial

Ex: a2 + b, x + y2 + 9 , y2+ xy + 8 . . . . etc

Cubic Polynomial

In a polynomial the largest exponent value of any given variable is two then it is called Quadratic polynomial

Ex: a3 + b, x3+ y2 + 9 , y3+ xy + 8 . . . . etc

Bi quadratic polynomial or Quartic polynomial

In a polynomial the largest exponent value of any given variable is four then it is calledQuartic polynomial

Ex: a4 + b3 + 2ab, x4 + y2+ 9 , 5y2 + x2y + 24 . . . . etc

Polynomial Function

A polynomial function is an expression constructed with one or more terms of variables with constant exponents. If there are real numbers denoted by a, then function with one variable and of degree n can be written as:

f(x) = a₀xⁿ + a₁x^n-1+ a₂x^n-2+ ….. + a_n-2x²+ a_n-1x + a_n

Degree of Polynomials

The degree of polynomials in one variable is the highest power of the variable in the algebraic expression.For a multivariable polynomial, it the highest sum of powers of different variables in any of the terms in the expression.

STANDARD FORM

A polynomial is in standard form when its term of highest degree is first, its term of 2nd highest is 2nd etc.. Examples of Polynomials inStandard Form. Non-Examples of Polynomials inStandard Form. x² + x + 3. 2y ⁴ + 3y ⁵ + 2+ 7.

Polynomial Division

If a polynomial has more than one term, we use long division method for the same. Following are the steps for it.

Write the polynomial in descending order.
Check the highest power and divide the terms by the same.
Use the answer in step 2 as the division symbol.
Now subtract it and carry down the next term.
Repeat step 2 to 4 until you have no more terms to carry down.
Note the final answer including remainder in the fraction form (last subtract term).

Division Algorithm:-

Dividend = Divisor x Quotient + Remainder or

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

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