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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

### Terms of Polynomials

Based on the number of terms in a polynomial, the polynomial is classified as monomial, binomial, etc.

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What is the word or phrase used to refer to parts of a polynomial consisting of product of numbers and variables?

• a term
• a term
• a coefficient

What is the word used to refer the number part in a term?

• a term
• a coefficient
• a coefficient

How many terms are there in the polynomial 3x^2y^5?

• 1
• 1
• 2
• 3

The answer is '1'. This is an example of polynomial with one term, called "monomial".

How many terms are there in the polynomial 3x^2 + y^5?

• 1
• 2
• 2
• 3

The answer is '2'. This is an example of polynomial with two terms, called "binomial"

How many terms are there in the polynomial x^2y+y^2z+xz^2?

• 1
• 2
• 3
• 3

The answer is '3'. This is an example of polynomial with 3 terms, called "trinomial"

Based on the number of terms in a polynomial the polynomials are classified as

•  monomial

•  binomial

•  trinomial

•  polynomial of n terms

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

• yes
• yes
• no

Is 3x^2 a polynomial?

• yes
• yes
• no

Consider the polynomials

•  (2x^2+x^2)

•  3x^2

Remember that the polynomials represent quantities. If x is a number and takes value 2, then

(2x^2+x^2) = 2 xx 2^2 + 2^2 = 12

3 xx 2^2 = 3 xx 4 = 12

Both of these two polynomials evaluate to identical values for all values of x.

Consider the polynomials

•  (2x^2+x^2)

•  3x^2

These polynomials are equal for all values of x. This is explained with the following.

2 x^2 + x^2
substitute multiplying by 2 as repeated addition.
 = x^2 + x^2 + x^2
count the repeated addition and convert to multiply by the number
 = 3 x^2
This is the second polynomial.

The above is generalized as distributive property of multiplication over addition. This part will be explained in "number system or algebra in high school".

For now, we can understand that two terms having same variable combination, the coefficients can be combined to form a single term.

Example: 2.3x^5y^7 + 1.1x^5y^7
quad = (2.3+1.1) x^5y^7
quad = 3.4x^5y^7

Terms that differ only in the coefficient part can be combined. Such terms are called "like terms".

Terms that differ in variables or exponents of variables are called "unlike terms".

Are 2x^3y^2 and (x^3y^2)/7 like or unlike terms?

• like terms
• like terms
• unlike terms

The answer is 'like terms'. The difference is in the coefficients 2 and 1/7.

Are 2x^2y and (x^2z)/7 like or unlike terms?

• like terms
• unlike terms
• unlike terms

The answer is 'unlike terms'. The terms differ in the variables.

Are 2xy^2 and 3.1(xy)^2 like or unlike terms?

• like terms
• unlike terms
• unlike terms

The answer is 'unlike terms'. The terms differ in the exponents of variables.

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

• a large water body
• having same characteristics or qualities
• having same characteristics or qualities

The answer is 'having same characteristics or qualities'.

What is two terms having identical variables with identical exponents called?

• Pronunciation : Say the answer once

What does the word 'unlike' mean?

• having different characteristics or qualities
• having different characteristics or qualities
• to remove water from a lake

The answer is 'having different characteristics or qualities'.

What are two terms having different variables or different exponents called?

• Pronunciation : Say the answer once

Like terms have same variables with same exponents.

Unlike terms differ in the variables or in the exponents.

Like Terms: The terms that has identical variables with identical exponents are called like terms.

Unlike Terms: The terms that differ in the variables or in the exponents of the variables are called unlike terms.

What type is the polynomial (2x^2+x^2) based on the number of terms?

• monomial
• monomial
• binomial
• trinomial

The answer is 'monomial'. The like terms can be combined and so the polynomial is a monomial.

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

• one term
• one term
• control over something

The answer is 'one term'. monomial means "one"(mono) + "term"(nomial).

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

• five terms
• two terms
• two terms

The answer is '2 terms'. binomial means "two"(bi) + "term"(nomial).

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

• three sides
• three terms
• three terms

The answer is 'three terms'. trinomial means "three"(tri) + "term"(nomial).

What is the term used to refer polynomial of 1 term ?

• Pronunciation : Say the answer once

What is the term used to refer polynomial of 2 terms ?

• Pronunciation : Say the answer once

What is the term used to refer polynomial of 3 terms ?

• Pronunciation : Say the answer once

Based on the number of terms in a polynomial the polynomials are classified as monomials, binomials, trinomials, and polynomials of n terms

Classification of polynomials based on the number of terms: After combining any like terms in polynomials, the number of terms is considered. Polynomials are classified as

•  monomial

•  binomial

•  trinomial

•  polynomial of n terms

Solved Exercise Problem:

What type is the polynomial (x^4+2xy^2+xy^2-x^4) based on the number of terms?

• monomial
• monomial
• binomial
• trinomial

The answer is 'monomial'. The like terms can be combined and two terms combine to 0. So, the polynomial is a monomial.

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