POLYNOMIALS

Learning Outcomes:

After reading the post the reader will be able to:

• What are the constant(مستقل مقدار), variable(متغیر), literal(حرفی مقدار) and algebraic expression. 
• Polynomial
• Degree of a polynomial(کثیر رقمی کا درجہ) 
• Coefficient(عددی سر) of a polynomial 
• Recognize polynomial in one, two and more variables(متغیرات).
• Recognize polynomials of various(مختلف) degrees (for example linear, quadratic, cubic and biquadratic polynomials).
• Add, subtract and multiply polynomials.
• Divide a polynomial by a linear polynomial.

ALGEBRAIC EXPRESSIONS: (الجبری جملے)

An algebraic expression is made up of symbols (علامات) and signs of algebra. Algebra helps us to make general formula (جنرل فارمولا) because algebra is linked with arithmetic. 

For example, x² + 2 x + 1 and √x -(1/√x) , "where x is not equal to zero"  are algebraic expressions.

Recall Constant, Variable, Literal (حرفی مقداریں)and Algebraic Expression.

Constant: A symbol that has a fixed numerical value (مخصوص عددی قیمت) is called a constant. 

For example in  2x+5, 2 and 7 is a constant term.

Variable: Variable is a symbol, usually a letter that is used to represent a quantity that may,have an infinite (uncountable) number of values are also called unknowns. For example, in 4x+ y + 3z; so x, y and z are variables.

Literal: The alphabets that are used to represent constants (مستقل مقدار) or coefficients are called literals. For example, in ax² + bx + c; so a, b and c are literals (حرفی مقداریں) whereas x is a variable.

Algebraic Expression: An expression which connects (جوڑنا) variables and constants by algebraic operations (الجبری عوامل) of addition, subtraction, multiplication and division is called an algebraic expression.

 A few algebraic expressions(الجبری جملے) are given below: 

(1) 24   (ii)  x + 2z   (iii) 4x -  y + 7  (iv) {- 2 /x}+ y   (v) 3y + 7z -6

Note:

[Algebra was introduced by Muslim Mathematician named Al Khawaraizmi(الخوارزمی) {780 - 850}. He was also considered as the father of modern Algebra] 

POLYNOMIAL: (کثیر رقمی)

Definition: A polynomial expression or simply a polynomial is an algebraic expression consisting(مشتمل) of one or more terms in each of which the exponent of the variable is zero(0) or a positive integer(مثبت صحیح عدد).

 For example, 13,  -1x,  5x + 2y,  x² — 3x + 6 are all polynomials. The following algebraic expressions(الجبری جملے) are not polynomials. 1/x , x⁻²,  x² - x⁻³ +5, x³ + y⁻⁴ , x/y.

Degree of a polynomial:(کثیر رقمی کا درجہ)

Degree of a Polynomial is the highest power(بڑی قوت) of a part (term) in a polynomial. Degree of a term in a polynomial is the sum of the exponents(قوت نما) on the variables in a single term. The degree of 2x³y⁴ is 7 as 3 + 4 = 7 

Coefficient of a Polynomial:

In a term the number multiplied by the variable (عددی سر) is the coefficient of the variable. In 4x + 6y, 4 is coefficient of x and 6 is coefficient of y.

Recognition (پہچان) of Polynomial in one, two and more Variables:

a) Polynomials in one Variable 

(ایک متغیر والی کثیر رقمی)

Consider the following Polynomials: 

(1) x⁴ + 4  (ii) x² + x + 1 (iii) y³ +y² - y + 1  (iv) y² - y + 8 , In these polynomials (i) and (ii) x is the variable and in polynomials (iii) and (iv) y is the variable. All these polynomials are polynomials in one(1)  variable.

b) Polynomials in two Variables:

(دو متغیر والی کثیر رقمی)

Consider the following Polynomials:

 (i) x² + y + 2 (ii) x²y + xy + 6 (iii) x²z + xz + z (iv) x²z + 8,  In polynomials (i) and (ii) x, y are the variables(متغیرات). In these above polynomials (iii) and (iv) x, z are the variables. All these polynomials(کثیررقوم) are in two variables.

c) Polynomials in more Variables:

 

Similarly x³yz + xy²z + xy + 7 is a polynomial in three variables x, y and z.

Recognition of polynomials of various degrees:

a) Linear Polynomials: (یک درجی کثیر رقمی)

Consider the following polynomials: 

(i)x + 2 (ii) x (iii) x + 2y (iv) x + z, In all these polynomials the degree of the variables x, y or z is one. Such types of polynomials are linear (یک درجی) polynomials.

b) Quadratic Polynomials: (دو درجی کثیر رقمی)

Let us write a few polynomials in which the highest exponent or sum of exponents is always 2. (i) x² (ii) x² - 3 (iii) xy + 1 

In the first two polynomials x is the variable and its degree(درجہ) is 2. In the 3rd Polynomial x, y are the variables and sum of their exponents is 1 + 1= 2. Its degree is also 2. Therefore above polynomials of the type (i), (ii) and (iii) are quadratic polynomials. 

c) Cubic polynomials:(سہ درجی کثیر رقمی)

Consider the following examples of polynomials:

(I) 5x³ + x² - 4x +1 

(ii) x²y + xy² + y - 3

The degree of each one of polynomial is 3. These polynomials are called cubic polynomials.

d) Biquadratic polynomials:(چہار درجی)

We take a few polynomials of 4 degree.

(i) x⁴ + x²y² + x³y + y³ + 3

(ii) y² + y⁴ + y³ + y +9

These polynomials are biquadratic(چہار درجی) polynomials.

Operations on polynomials:

Addition of polynomials ( algebraic expressions)

If P(x) and Q(x) are polynomials then their addition is represented by P(x) + Q(x). In order to add two or more than two polynomials first we write the polynomials in ascending or descending order and like terms each in the form of columns. Finally we add the coefficients of like terms.

Example: Find the sum of ( 3x³ + 5x² - 4x ) , ( x³ - 6 + 3x² ) and ( 6 - x² - x )

Solution:

Subtraction of polynomials ( algebraic expressions)

The subtraction of polynomials P(x) and Q(x) is represented as P(x) - Q(x). If the sum of two polynomials is zero then P(x) and Q(x) are called additive Inverse of each other.

Same as addition method we write the polynomials in ascending or descending order and then we change the sign of every term of the polynomial which is to be subtracted.

Example: solve ( 5x⁴ + x - 3x² - 9 ) - ( 2x³ - 4x² + 8 - x )

Solution:

Multiplication of polynomials:

Multiplication of the polynomials is explained by following examples

Example: Solve ( 4x² ) × ( 5x³ )

Solution:

Example: Find product of following polynomials.

( 3x² + 2x - 4 ) and ( 5x² - 3x + 3 )

Solution:

Example: Solve ( 2x - 3 ) × ( 5x + 6 )

Solution:

Division of polynomials:

The division is reverse process of Multiplication. The division of polynomials is explained by following examples.

Example: ( -8x⁵ ) ÷ ( -4x³ )

Solution:

Example: ( x³ - 2x + 4 ) ÷ ( x + 2 )

Solution:

Note that if a polynomial is completely (exactly) divisible by another polynomial then the remainder is zero (0).

Overall Summary:

An expression (sentence) which Connects or links variables and constants by algebraic operations of addition subtraction, multiplication and division is called an algebraic expression. 

Constants are algebraic symbols that have a fixed value and these values are unchange able.

A symbol(sign) in algebra which can assume different numerical values is called a variable. 

A literal (حرفی مقدار) is a value that is expressed as itself. For example, the number 45 or the word "speed" are both literals. 

An algebraic expression which has finite (countable ) number of terms and the exponents of variables are whole numbers, is called polynomial. 

A polynomial is either zero (0) or an be written as the sum of a finite number of non-zero terms.

In a polynomial coefficient is a number or symbol(sign) multiplied with a variable in an algebraic term.

 The polynomials of degree one(1) are called linear polynomials. 

The polynomials of degree two(2) are called quadratic polynomials. 

The polynomials of degree three(3) are called cubic polynomials. 

The polynomials of degree four(4) are called biquadratic polynomial.

Have a nice day.