Operations With Polynomials

Polynomials


A polynomial is a mathematical expression that is constructed from one or more variables and constants, using only the operations of addition, subtraction, and multiplication. The constants of the polynomials are real numbers, whereas the exponents of the variables are positive integers.

For example, ` x^2 + 5x-3` is a polynomial in a single variable x.


Degree of a Polynomial:


The highest power of the variable present in a polynomial is called the degree of the polynomial. In the above example, the degree of the polynomial is 2.


Coefficients of a Polynomial:


The constant values present in a polynomial are knows as its coefficients / coefficient values. The constants used in the above polynomial are 1, 5 and -3.


Variables of a Polynomial:


Variables are alphabets like a,b,c, x, y, z etc that are used in a polynomial. They are called variables because they can take up any value from a given range (thus called "vary-ables"). In the above example, x is the variable. There also exist polynomials that use more than one variable. For example,` x^2 + 5xy-3y^2 ` is a polynomial of degree 2 in two variables x and y.


Note:

Until this point, we have only mentioned what a polynomial is. However, for many reasons it is wise to make clear as to what is not a polynomial. Without this specification, it is likely that you render a non-polynomial to be a polynomial.


Caution: What is not a polynomial?

  •  A polynomial cannot have negative values of exponents. E.g. 4x-2 can not be a polynomial term.
  •  A polynomial term can not have the variable in the denominator. E.g. 4/x is not a polynomial term.
  •  A polynomial term cannot have a variable inside the radical sign. 3xsqrtx is therefore not a polynomial term.

Each part of a polynomial that is being added or subtracted is called a "term". So the polynomial has three terms.


Polynomial Terms

Shown above is a simple example of the polynomial, and this is how polynomials are usually expressed. The term having the largest value of exponent (2 in this case) is written first, and is followed by the term with the next lower value of exponent which in turn is followed by a term with the next lower exponent value and so on. The term with the maximum value of exponent is called the "Leading Term" and the value of its exponent is called the "degree of the polynomial".


Polynomial Degree Examples


Example 1:
 6

Explanation:

Since the highest power of the variable present in a polynomial is called the degree of the polynomial,

And

The given polynomial has four terms with the term 5x6 having the largest value of exponent equal to 6,


Example 2:
 4

Explanation:

The given polynomial involves two variables x and y. And each term in the polynomial has some degree of each of the two variables. In this case,

The degree of each term is given by the sum of the exponents of the two variables involved in the term, and the degree of the entire polynomial is given by the maximum value of the sum.

In order to determine the degree of the polynomial, we calculate the degree of individual terms first:

Degree of first term ` (4x^3y): 3 + 1 = 4 `

Degree of second term `(6x^2y): 2 + 1 = 3 `

Degree of third term `(4xy): 1 + 1 = 2 `

Degree of fourth term `(3y^2): 0 + 2 = 2 `


Point to Ponder


When a variable is absent in a term, its exponent is zero)

Now the degree of the polynomial is given by the maximum of the above degrees.

=> Degree of the given polynomial = max( 4, 3, 2, 2)

=> Degree of the given polynomial = 4

( Interesting Note: 0 is also a polynomial)


Classification of Polynomials on the basis of their Degrees

Polynomials are very often classified depending on their corresponding degrees. You must have a brief idea of what is a linear or quadratic polynomial.

  •  A polynomial of degree = 1 such as 3x + 5 is called 'linear'
  •  A second-degree polynomial such as 5x2 + 3x-2 is called 'quadratic'
  •  A third-degree polynomial is referred to as a 'cubic'. E.g. ` x^3-6x^2 + 2 ` is a cubic
  •  A polynomial of degree = 4 is sometimes called a 'quartic' or 'biquadratic'

Similarly we can refer to other higher degree polynomials as 'quintic' , 'hexic' etc.


Some Basic Properties of Polynomials

Polynomials have a few basic properties that are peculiar to them. These have been presented below:


  •  The sum of 2 or more polynomials gives another polynomial.
  •  A product of 2 or more polynomials also gives a polynomial.
  •  The derivative of a polynomial with degree n produces another polynomial with a degree n-1
  •  Polynomials can be factorized into the product of two or lower degree polynomials.

Next

We learn some basic examples related with polynomials that will help you have a thorough understanding of polynomials and their related concepts.


Simple Ploynomial Examples


Example 1:

Explanation:

Each term in a polynomial is a combination of a number (positive or negative) and a variable. The numbers involved in a polynomial are termed as its coefficients. There are 4 terms in the given polynomial, hence there are four coefficients as well. These coefficients are 5, 6,-3,-1. Some people however do not take the constant term as a coefficient.


Example 2:

Explanation:

The exponents are the powers of variables present in a polynomial. In the given polynomial, exponents are 3, 2, 1 and 0 Whenever there is no power mentioned on a variable, its power is equal to one as shown below.

Exponents in Polynomials

Example 3:
 88

Explanation:

In order to evaluate a polynomial at certain value/values of the variables involved, we simply put in the value of that variable / those variables in to the polynomial and do the ordinary addition, subtraction and multiplications present in the polynomial expression.

Since the given polynomial involves a single variable x, we replace x with 3 in the polynomial.

= 54 + 45-21 + 10

= 88

Hence the given polynomial evaluates to 88 at x = 3.


Example 4:
 24

Explanation:

The given polynomial involves two variables x and y, and in order to evaluate the polynomial for given values of x and y, we simply replace x with 1 and y with 2.

=> ` 4(1)^3 (2) + 6(1)^2 (2)-4(1)(2) + 3(2)^2 `

= ` 4(1)(2) + 6(1)(2)-4(1)(2) + 3(4)`

= ` 8 + 12-8 + 12`

= 24

Hence the given polynomial evaluates to 24 at x = 1, y = 2


Example 5:

Examples:

 `3x-8`

`3x-8 ` is linear (degree 1)


  `x + 7 `

`x + 7 ` is linear (degree 1)


  `5x^2 -4x + 3 `

`5x^2 -4x + 3 ` is quadratic (degree 2)


  `x^3 + x2-1 `

`x^3 + x^2-1 ` is cubic (degree 3)


 `x^2 -4 `

`x^2 -4 ` is quadratic (degree 2)


  `2y^5-6y^4 + 12y^3 + y^2 + 4 `

`2y^5-6y^4 + 12y^3 + y^2 + 4 ` is quintic (degree 5)


  `7x^4 -4 `

`7x4 -4 ` is quartic / biquadratic (degree 4)



Some Easy-to-Learn Properties of Algebra

Let a,b,c be three variables. Then the followings are some basic rules of algebra applicable to these variables.


 Commutative Law For Addition

` a+ b = b + a `


 Commutative Law For Multiplication

`a*b=b*a`


 Associative Law For Addition

` a+( b+c ) = ( a+b )+c`


 Associative Law For Multiplication

` a*( b*c )=( a*b )*c`


 Distributive Law

`a*( b+c ) = ( a*b )+( a*c )`


 Cancellation Law For Addition

` a+c=b+c iff a=b `


 Cancellation Law For Multiplication

`a*c=b*c iff a=b` where `c!=0`



In the next section, we make use of these simple formulas to find solutions to given problems.



Example 6:
 5

Explanation:

Making use of the Commutative Law For Addition stated above

a + b = b + a

=> For the given expression, we have x = 5


Example 7:
 2

Explanation:

Making use of the Commutative Law For Multiplication stated above

a . b = b . a

=> For the given expression, we have r = 2


Example 8:
 2

Explanation:

According to the Distributive Law, we have

`a*( b + c ) = ( a*b ) + ( a*c )`

Through comparison of the given expression with the above formula, we conclude that

m=a=2

We can also prove that m = 2 by method of equating the two sides of the given expression as shown below.

Given that

  • 2 . ( 3 + 1 ) = ( 2 . 3 ) + ( m . 1 )
  • L.H.S (Left Hand Side)

    = 2 . ( 3 + 1 )
  • = 2 . 4
  • = 8
  • R.H.S (Right Hand Side)

    = ( 2 . 3 ) + ( m . 1 )
  • = 6 + m
  • By equating the two sides,

  • 8 = 6 + m
  • m = 2 (Subtract 6 from both sides of the equation)

Hence we get the same answer as we got using the Distributive Formula.


Example 9:
 5

Explanation:

Here we make use of the Associative Law For Multiplication which states that for any variables a, b and c

a . ( b . c ) = ( a . b ) . c

Through comparison we get

d = c = 5


Example 10:
 5

Explanation:

Using the Cancellation law For Addition

`a + c = b + c iff a = b`

Here

a = x

b = 5

c = 2

And the formula states that the identity will hold if and only if a = b i.e., if x = 5


Example 11:
 5

Explanation:

Using the Cancellation law For Multiplication

`a*c = b*c iff a=b ` where `c!= 0`

Here

a = n

b = 5

c = 3 `!=` 0

Hence the equation will hold if and only if a = b, that is if n = 5


Example 12:
 56

Explanation:

We have studied Distributive law which states that

` a*( b + c )=( a*b ) + ( a*c ) `

Here we can split up 14 = 10 + 4

=> 4 . ( 10 + 4 ) = ( 4 . 10 ) + ( 4 . 4 )

=> 4 x 14 = 40 + 16 = 56

Hence we can split our problem using this property and find the solution with greater ease 


Example 13:
 3 ( x + 3 )

Explanation:

We have studied that in the distributive property, we distribute the operation of multiplication over addition. We can use this property to simplify the above expression.

We see that both 3, and 9 are multiples of 3. So if we take 3 out of the above expression, leaving the rest inside the parentheses, what we get is a distributed equivalent of the above expression.

=> 3x + 9 = 3 ( x + 3 )

A mentionable point here is that the property also holds for subtraction. For example

=> 3x-9=3 ( x-3 )=3 ( x+(-3 ) )

is also true.


Example 14:
 8 x n

Explanation:

Using the Associative Property for Multiplication

` a*( b*c ) = ( a*b )*c `

=> 2 x ( 4 x n ) = ( 2 x 4 ) x n

=> 2 x ( 4 x n ) = 8 x n



Non-Commutative Operations


Here we need to mention that subtraction, division, matrix multiplication, vector product are all non-commutative.

For Example :
 `4-2!=2-4` => Subtraction is non-commutative
 `( 4 )/( 2 )!=2/( 4 )` => Division is non-commutative

Similarly Matrix Multiplication and vector Product can be shown to be non-commutative.







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