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Polynomial Arithmetics

Polynomial Arithmetics

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Simplification of Polynomials


 »  numbers in different forms
    →  simplified form of `7-2/2` is `6`
    →  prime factorization of `6` is `3 xx 2`


 »  Polynomials are simplified to General form: sum of irreducible terms
irreducible terms are terms that cannot be equivalently given as multiple simpler terms.
    →  eg: `x^3-x^2-4x+4`


 »  Polynomials in factors form: product of irreducible factors.
    →  eg: `(x-1)(x-2)(x+2)`

Simplification of Polynomials

plain and simple summary

nub

plain and simple summary

nub

dummy

polynomial simplification can be in one of the two forms

 •  general form of polynomial

 •  factors form of polynomial

simple steps to build the foundation

trek

simple steps to build the foundation

trek

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Polynomials can be equivalently represented in different forms. The simplified polynomial form is given as sum of terms that are reduced to multiplication of variables. The factors form is given as product of factors that are reduced to the smallest factors.


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Starting on learning "Simplification of Polynomials". ;; Polynomials can be equivalently represented in different forms. The simplified polynomial form is given as sum of terms that are reduced to multiplication of variables. The factors form is given as product of factors that are reduced to the smallest factors.

Which of the following equals `6`?

  • `6`
  • `3+2(4-2)-1`
  • `3xx2`
  • all the above

The answer is 'all the above'

All the following represent the same quantity.

 » `6` : simplified form

 » `3+2(4-2)-1`

 » `3xx2` : prime factors form

A numerical expression may be evaluated to a number and that is "simplified form"

A number may be represented as product of prime factors and that is "prime factors form"

Which of the following equals `x^2+x(x+4)-30`?

  • `2x^2+4x-30`
  • `12x^2-x^2(7+3)`` -26+4x-2^2`
  • `(2x-6)(x+5)`
  • all these are equal

The answer is 'all these are equal'

An expression can be represented in multiple forms.

 »  `x^2+x(x+4)-30`

 »  `2x^2+4x-30`

 »  `12x^2-x^2(7+3)`` -26+4x-2^2`

 »  `(2x-6)(x+5)`

In these, two forms are simplified.

 »  `2x^2+4x-30` : Simplified Polynomial in general form which is given as addition or subtraction of irreducible terms.

 »  `(2x-6)(x+5)` : Polynomial in factors form, which is given as product of irreducible factors.

Given polynomial `x^2+ x(x+2)-23`, What is the part `x(x+2)` called?

  • a term
  • a sub-expression

The answer is 'a sub-expression'. `x(x+2)` is not a term as it can be given as `x^2+2x` which has two terms.

The given polynomial `x^2 + x(x+2)-23` can be equivalently given as follows.

`x^2 + x(x+2)-23`
`=x^2 + x^2+2x-23`
`=2x^2 +2x-23`

The derived polynomial `2x^2 +2x-23` is simpler and has terms that cannot be reduced further. This polynomial has irreducible terms.

Given polynomial `(x-1)(x^2-4)`, What is the part `(x-1)` called?

  • a factor
  • a term

The answer is 'a factor'.

The given polynomial `(x-1)(x^2-4)`can be equivalently given as follows.

`(x-1)(x^2-4)`
`=(x-1)(x-2)(x+2)`

The derived polynomial `(x-1)(x-2)(x+2)` is simpler and has factors that cannot be factorized further. This polynomial form is in irreducible factors.

comprehensive information for quick review

Jogger

comprehensive information for quick review

Jogger

dummy

Polynomial Simplification: A polynomial can be simplified to

 •  general form: sum of terms reduced to exponent of variables without any addition or subtraction.

 •  factors form: product of polynomials factorized to irreducible factors.



           

practice questions to master the knowledge

Exercise

practice questions to master the knowledge

Exercise

Express `(x+2)(2x-2)` in general form of polynomial.

  • `2x^2+2x-4`
  • `x+4x-2`
  • `5x-2`

The answer is '`2x^2+2x-4`'

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Which of the following equals 6 ?
6
6
plus;minus
3 plus 2 times 4 minus 2, minus 1
multiplied
3 multiplied 2
all;above
all the above
The answer is 'all the above'
All the following represent the same quantity. 6 : simplified form. 3 plus 2 times 4 minus 2 minus 1. ;; 3 multiplied 2 : prime factors form. ;; A numerical expression may be evaluated to a number and that is "simplified form". ;; A number may be represented as product of prime factors and that is "prime factors form"
Which of the following equals x squared + x multiplied x + 4 minus 30.
1
2
3
4
The answer is 'all these are equal'
An expression can be represented in multiple forms. In these two forms are simplified.;; 2 x squared + 4 x minus 30: simplified polynomial in general form which is given as addition or subtraction of irreducible terms. ;; 2x minus 6 multiplied x + 5 : Simplified Polynomial in factors form, which is given as product of irreducible factors.
Given polynomial, what is the part x multiplied x + 2 called?
term
a term
sub;expression;express
a sub-expression
The answer is "a sub-expression". x multiplies x+2 is not a term as it can be given as x squared + 2 x which has two terms.
The given polynomial can be equivalently reduced. ;; The derived polynomial 2 x squared + 2 x minus 23 is simpler and has terms that cannot be reduced further. This polynomial has irreducible terms.
Given polynomial, what is the part x minus 1 called?
factor
a factor
term
a term
The answer is 'a factor'.
The given polynomial can be equivalently factorized. ;; The derived polynomial is simpler and has factors that cannot be factorized further. This polynomial form is in irreducible factors
polynomial simplification can be in one of the two forms general form of polynomial and factors form of polynomial
Polynomial Simplification : A polynomial can be simplified to;; general form: sum of terms reduced to exponent of variables without any addition or subtraction. ;; factors form: product of polynomials factorized to irreducible factors.
express x+2 multiplied 2x minis 2 in general form of polynomial.
1
2
3
The answer is "2 x squared +2 x minus 4"

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