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Thought-Process to Discover Knowledge

Welcome to nubtrek.

Books and other education websites provide "matter-of-fact" knowledge. Instead, nubtrek provides a thought-process to discover knowledge.

In each of the topic, the outline of the thought-process, for that topic, is provided for learners and educators.

Read in the blogs more about the unique learning experience at nubtrek.
mathsIntroduction to Algebra, Polynomials, and IdentitiesPolynomials - Basics

### Introduction to Polynomials

In this page, polynomials are introduced as simplified form of algebraic expressions.

click on the content to continue..

What is an algebraic expression?

• a statement of quantity as sum of terms of numbers and variables
• a statement of quantity as sum of terms of numbers and variables
• sequence of numbers having a pattern

The answer is 'a statement of quantity as sum of terms of numbers and variables'

Is 3x+5 an algebraic expression?

• yes
• yes
• no

"Water is cold." Is that an algebraic expression?

• yes
• no
• no

Is 7sqrt(x)+5 an algebraic expression?

• yes
• yes
• no

Is 7 y+5 an algebraic expression?

• yes
• yes
• no

Note the two algebraic expressions

•  7sqrt(x)+5

•  7y+5

The two expressions share some similarities. In the first, the variable quantity is denoted with x. If the square root of the same variable quantity is denoted with y, (ie y=sqrt(x)) then the second expression is arrived at.

7sqrt(x)+5 has a fraction as exponent, i.e. x^(1/2). But 7y+5 does not have a fraction as exponent. These two expressions are similar, with the quantity that is represented by variable is in different forms. If x represents length of an iron bar, then y represents the square root of the length of the iron bar. By making a different choice on the variable representing the quantity, the expressions are arrived at. If 7sqrt(x)+5 is given, an user can just change the meaning of variable and take the expression as 7y+5.

An expression with fraction as exponent can be equivalently represented with an expression with whole number as exponents.

Is 5x^(-2)+3 an algebraic expression?

• yes
• yes
• no

Is (5+3x^2)/(x^2) an algebraic expression?

• yes
• yes
• no

Note the two algebraic expressions

•  5x^(-2)+3

•  (5+3x^2)/(x^2)

The two expressions share some similarities. In the first, the variable x has negative exponent. In the second, the quantity is given as division of two expressions with positive exponents.

Note that the two expressions are equivalent.

An expression with exponents of fraction or negative integers can equivalently be given as expressions with exponents of whole numbers.

We are learning methods to analyze algebraic expressions and if the set of expressions can be simplified, then the theory to analyze them is simpler.

The polynomials are simplified subset of algebraic expressions and have only whole numbers as exponent of variables.
Example: 4x^3+1 is a polynomial

But (4x^3+1)/(x^2) is not a polynomial. This expression is given as division of two polynomials.

7x^(1/5)-4 is not a polynomial as the variable has exponent 1/5, a fraction.

3x^(-3)+x^2 is not a polynomial as the variable has exponent -3, a negative integer.

Note: the numbers or coefficients in polynomials can have negative integer or fraction as exponents. The following are all polynomials

(4x^3+1)/(2^2) division by a number

7^(1/5)x^2-4fraction as exponent for a number

3^(-3)x^5+x^2negative number as exponent for a number

Is 7sqrt(x)+5 a polynomial?

• yes
• no
• no

The answer is 'No'. The variables has to have whole numbers as exponents.

Is 7y+5 a polynomial?

• yes
• yes
• no

Is 7^(1/8) y+sqrt(5) a polynomial?

• yes
• yes
• no

The answer is 'Yes'. The coefficients can have fractions as exponents.

Is 5x^(-2)+3 a polynomial?

• yes
• no
• no

The answer is 'no'. The variables has to have whole numbers as exponents.

Is (5+3x^2)/(x^2) a polynomial?

• yes
• no
• no

The answer is 'no'. The whole expression is not a polynomial as it involves division. Consider,
(5+3x^2)/(x^2)
quad = (5+3x^2) xx x^(-2)
This is not a polynomial.

Is (5+3x^2) a polynomial?

• yes
• yes
• no

Is (5^(-3)+3x^2) a polynomial?

• yes
• yes
• no

The answer is 'Yes'. Even when the coefficient has power -3, it is a polynomial.

Which of the following is a meaning for the word 'polynomial'?

• many terms; multi parts
• many terms; multi parts
• several islands

The answer is 'many terms; multi parts'.

What is the term used to refer algebraic expressions with whole numbers for exponents?

• Pronunciation : Say the answer once

Polynomials : An algebraic expression with whole numbers as exponents of variables is a polynomial.

Solved Exercise Problem:

Is sqrt(2)xy+y^2+ x^3 a polynomial?

• yes
• yes
• no

The answer is 'Yes'. The coefficients can have fractions as exponents.

Solved Exercise Problem:

Is xz^(-2)+3 a polynomial?

• yes
• no
• no

The answer is 'no'. The variables has to have whole numbers as exponents.

Solved Exercise Problem:

Is (x^3+y^3+z^3+p^3+q^3) a polynomial?

• yes
• yes
• no

The answer is 'Yes'. This has 5 variables, x, y, z, p, q.

Solved Exercise Problem:

Is (7^(-3)+3^(-2)x^2)` a polynomial?

• yes
• yes
• no

The answer is 'Yes'. The variable has whole number as exponent. The numbers can take any negative exponents.

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