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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.

User Guide

Welcome to nubtrek.

The content is presented in small-focused learning units to enable you to
think,
figure-out, &
learn.

Just keep tapping (or clicking) on the content to continue in the trail and learn.

User Guide

To make best use of nubtrek, understand what is available.

nubtrek is designed to explain mathematics and science for young readers. Every topic consists of four sections.

nub,

trek,

jogger,

exercise.

User Guide

nub is the simple explanation of the concept.

This captures the small-core of concept in simple-plain English. The objective is to make the learner to think about.

User Guide

trek is the step by step exploration of the concept.

Trekking is bit hard, requiring one to sweat and exert. The benefits of taking the steps are awesome. In the trek, concepts are explained with exploratory questions and your thinking process is honed step by step.

User Guide

jogger provides the complete mathematical definition of the concepts.

This captures the essence of learning and helps one to review at a later point. The reference is available in pdf document too. This is designed to be viewed in a smart-phone screen.

User Guide

exercise provides practice problems to become fluent in the concepts.

This part does not have much content as of now. Over time, when resources are available, this section will have curated and exam-prep focused questions to test your knowledge.

summary of this topic

### Polynomials - Basics

Voice

Voice

Home

Introduction to Polynomials

»  Equivalence in algebraic expressions: algebraic expressions can be equivalently given with the exponent of variables in whole numbers.

→  eg 1: x^2+ sqrt(x)-3 is equivalently y^4+y-3 where y^2=x

→  eg 2: x^(-2)-x+1 is equivalently (1-x^3+x^2)/(x^2) which is ratio of two simplified expressions

»  Polynomials are simplified algebraic expressions with exponents of variables as whole numbers.

the algebraic expressions are used in equations and are analyzed (eg: finding factors or poles). The simplification into polynomials helps to formalize the theory and study the expressions.

### Introduction to Polynomials

plain and simple summary

nub

plain and simple summary

nub

dummy

Polynomials are simplified algebraic expressions with exponents of variables as whole numbers.

simple steps to build the foundation

trek

simple steps to build the foundation

trek

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

Keep tapping on the content to continue learning.
Starting on learning "Introduction to Polynomials". ;; In this page, polynomials are introduced as simplified form of algebraic expressions.

What is an algebraic expression?

• 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
• no

The answer is 'yes'

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

• yes
• no

The answer is 'no'

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

• yes
• no

The answer is 'Yes'

Is 7 y+5 an algebraic expression?

• yes
• no

The answer is 'Yes'

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.

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
• no

The answer is 'Yes'

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

• yes
• no

The answer is 'Yes'

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 whole number exponents.

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)

7^(1/5)x^2-4

3^(-3)x^5+x^2

Is 7sqrt(x)+5 a polynomial?

• yes
• no

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

Is 7y+5 a polynomial?

• yes
• no

The answer is 'Yes'

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

• yes
• no

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

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

• yes
• no

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

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

• yes
• 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
• no

The answer is 'Yes'

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

• 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
• several islands

The answer is 'many terms; multi parts'.

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

• Practice Saying the Answer

The answer is 'polynomial'.

comprehensive information for quick review

Jogger

comprehensive information for quick review

Jogger

dummy

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

practice questions to master the knowledge

Exercise

practice questions to master the knowledge

Exercise

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

• yes
• no

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

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

• yes
• no

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

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

• yes
• no

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

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

• yes
• no

The answer is 'Yes'. Even when the coefficient has power -3, the variable has whole number as exponent.

Progress

Progress

What is an algebraic expression?
statement;quantity;sum;terms
a statement of quantity as sum of terms of numbers and variables
sequence;numbers;pattern
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;s
yes
no
no
The answer is 'yes'
"Water is cold." Is that an algebraic expression?
yes;s
yes
no
no
The answer is 'no'
Is 7, square root of x + 5 an algebraic expression?
yes;s
yes
no
no
The answer is 'Yes'
Is 7 y+5 an algebraic expression?
yes;s
yes
no
no
The answer is 'Yes'
Note the two algebraic expressions. 7 square root x + 5 and 7 y + 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, then the second expression is arrived at. ;; An expression with fraction as exponent can be equivalently represented with an expression with whole number as exponents.
Is 5 x power minus 2 + 3 an algebraic expression?
yes;s
yes
no
no
The answer is 'Yes'
Is 5 + 3 x power 2 divided by x squared an algebraic expression?
yes;s
yes
no
no
The answer is 'Yes'
Note the two algebraic expressions. 5 x power minus 2 + 3, and 5 + 3 x power 2 divided by x squared ; 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, whole number exponents. ;; 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: 4 x cube + 1 is a polynomial. But 4 x cube + 1 divided by x squared is not a polynomial. This expression is given as division of two polynomials. ;; 7 x power 1 by 5 minus 4 is not a polynomial as the variable has exponent 1 by 5 : a fraction. ;; 3 x power minus 3 plus x squared is not a polynomial as the variable has exponent minus 3: a negative integer.
the numbers or coefficients in polynomials can have negative integer or fraction as exponents. The following are all polynomials having division by a number, fraction exponent for a number, and negative exponent for a number.
Is 7 square root of x + 5 a polynomial?
yes;s
yes
no
no
The answer is 'No'. The variables has to have whole numbers as exponents.
Is 7y+5 a polynomial?
yes;s
yes
no
no
The answer is 'Yes'
is 7 power 1 by 8 y + square root 5 a polynomial?
yes;s
yes
no
no
The answer is 'Yes'. The coefficients can have fractions as exponents.
Is 5 x power minus 2 + 3 a polynomial?
yes;s
yes
no
no
The answer is 'no'. The variables has to have whole numbers as exponents.
Is 5 + 3 x power 2 divided by x squared a polynomial?
yes;s
yes
no
no
The answer is 'no'. The whole expression is not a polynomial, as it involves division.
Is 5 + 3 x power 2 a polynomial?
yes;s
yes
no
no
The answer is 'Yes'
Is 5 power minus 3 + 3 x power 2 a polynomial?
yes;s
yes
no
no
The answer is yes. Even when the coefficient has power minus 3, it is a polynomial.
Which of the following is a meaning for the word 'polynomial'?
many;term;multi;parts
many terms; multi parts
several;islands
several islands
The answer is 'many terms; multi parts'.
What is the term used to refer algebraic expressions with whole numbers for exponents?
polynomial
The answer is 'polynomial'.
Polynomials are simplified algebraic expressions with exponents of variables as whole numbers.
Polynomials : An algebraic expression with whole numbers as exponents of variables is a polynomial.
Is square root 2 x y + y squared + x cube a polynomial?
yes;s
yes
no
no
The answer is 'Yes'. The coefficients can have fractions as exponents.
Is xz power minus 2 + 3 a polynomial?
yes;s
yes
no
no
The answer is 'no'. The variables has to have whole numbers as exponents.
Is the given algebraic expression a polynomial?
yes;s
yes
no
no
The answer is 'Yes'. This has 5 variables, x , y , z , p , q .
Is 7 power minus 3 + 3 power minus 2 x power 2 a polynomial?
yes;s
yes
no
no
The answer is yes. Even when the coefficient has power minus 3, the variable has whole number as exponent.

we are not perfect yet...