Server Not Reachable. *This may be due to your internet connection or the nubtrek server is offline.*

Thought-Process to Discover Knowledge

Welcome to **nub****trek**.

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.continue

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.continue

Welcome to **nub****trek**.

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. continue

The content is presented in small-focused learning units to enable you to

think,

figure-out, &

learn.

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.

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

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. continue

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. continue

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. continue

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. *

*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

You are learning the free content, however do shake hands with a coffee to show appreciation.

*To stop this message from appearing, please choose an option and make a payment.*

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

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.

*your progress details*

Progress

*About you*

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.