__maths__>__Introduction to Algebra, Polynomials, and Identities__>__Polynomial Arithmetics__### 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.

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Which of the following equals `6`?

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

The answer is 'all the above'

All the following represent the same quantity.

» `6` : simplified form

» `3+2(4-2)-1` : One of many numerical expression forms

» `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
- 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
- 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 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*.

polynomial simplification can be in one of the two forms

• general form of polynomial

• 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.

*Solved Exercise Problem: *

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

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

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

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