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Basic Algebraic Identities

Basic Algebraic Identities

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Algebraic Identities of Cube


 »  `(x+y)^3 ``= x^3+3x^2y+3xy^2+y^3`


 »  `(x-y)^3 ``= x^3-3x^2y+3xy^2-y^3`


 »  `x^3+y^3 ``= (x+y)(x^2-xy+y^2)`


 »  `x^3-y^3 ``= (x-y)(x^2+xy+y^2)`

Algebraic Identities of Cube

plain and simple summary

nub

plain and simple summary

nub

dummy

Algebraic identities of cube are derived using identities of squares and multiplication of polynomials.

simple steps to build the foundation

trek

simple steps to build the foundation

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In this page, the algebraic identities involving exponent 3 are introduced and explained.


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Starting on learning "Algebraic Identities of Cube". ;;In this page, the algebraic identities involving exponent 3 are introduced and explained.

Which of the following is equivalent to `(a+b)^3`?

  • `(a+b)(a+b)^2`
  • `a+b^3`
  • `a^3+b^3`

The answer is '`(a+b)(a+b)^2`'

`(a+b)^3`

`= color(coral)((a+b))color(deepskyblue)((a+b)^2)`

`= color(coral)((a+b))color(deepskyblue)((a^2+2ab+b^2))`

`= color(coral)((a+b))color(deepskyblue)(a^2) ``+ color(coral)((a+b))color(deepskyblue)(2ab)``+ color(coral)((a+b))color(deepskyblue)(b^2)`

`= color(coral)(a)color(deepskyblue)(a^2) ``+ color(coral)(b)color(deepskyblue)(a^2) ``+ color(coral)(a)color(deepskyblue)(2ab)``+ color(coral)(b)color(deepskyblue)(2ab)``+ color(coral)(a)color(deepskyblue)(b^2)``+ color(coral)(b)color(deepskyblue)(b^2)`

`= a^3+ 3a^2b + 3ab^2+b^3`

Which of the following is equivalent to `(a-b)^3`?

  • `(a-b)(a-b)^2`
  • `a-b^3`
  • `a^3-b^3`

The answer is '`(a-b)(a-b)^2`'

`(a-b)^3`

`= color(coral)((a-b))color(deepskyblue)((a-b)^2)`

`= color(coral)((a-b))color(deepskyblue)((a^2-2ab+b^2))`

`= color(coral)((a-b))color(deepskyblue)(a^2) ``- color(coral)((a-b))color(deepskyblue)(2ab) ``+ color(coral)((a-b))color(deepskyblue)(b^2)`

`= color(coral)(a)color(deepskyblue)(a^2) - color(coral)(b)color(deepskyblue)(a^2) ``- color(coral)(a)color(deepskyblue)(2ab)``-(- color(coral)(b)color(deepskyblue)(2ab))``+ color(coral)(a)color(deepskyblue)(b^2)``- color(coral)(b)color(deepskyblue)(b^2)`

`= a^3 - 3a^2b +3a b^2 - b^3`

Comparing the two identities

`(a_1+b_2)^3 ``= a_1^2+3a_1^2b_1``+3a_1 b_1^2 + b_1^3`

`(a_2-b_2)^3 ``= a_2^2-3a_1^2b_1``+3a_1 b_1^2 - b_1^3`

Substituting in the first identity `b_1 =- b_2`; the second identity is derived. Students may work this out to understand.

An identity for `a^3+b^3` is derived.

Upon substituting `a=-b`, the expression becomes `0`. So, `a+b` is a factor. The polynomial division principles are applied to arrive at. `a^3+b^3``=(a+b)(a^2-ab+b^2)`

An identity for `a^3-b^3` is derived.

Upon substituting `a=b`, the expression becomes `0`. So, `a-b` is a factor. The polynomial division principles are applied to arrive at.
`a^3-b^3``=(a-b)(a^2+ab+b^2)`

Note: The same can be derived using `a^3+c^3` by substituting `c=-b`.

Comparing the two identities

`a_1^3-b_1^3``=(a_1-b_1)``(a_1^2+a_1b_1+b_1^2)`
`a_2^3-b_2^3``=(a_2-b_2)``(a_2^2+a_2b_2+b_2^2)`

Substituting in the first identity `b_1 =- b_2`; the second identity is derived. Students may work this out to understand.

comprehensive information for quick review

Jogger

comprehensive information for quick review

Jogger

dummy

Algebraic Identities of Cube :
`(x+y)^3 ``= x^3+3x^2y+3xy^2+y^3`

`(x-y)^3 ``= x^3-3x^2y+3xy^2-y^3`

`x^3+y^3 ``= (x+y)(x^2-xy+y^2)`

`x^3-y^3 ``= (x-y)(x^2+xy+y^2)`



           

practice questions to master the knowledge

Exercise

practice questions to master the knowledge

Exercise

Which of the following helps to find `(a+b)^4`?

  • `(a+b)^2 xx (a+b)^2`
  • `(a+b) xx (a+b)^3`
  • both the above

The answer is 'both the above'

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Progress

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Progress

Which of the following is equivalent to a + b whole cube?
1
2
3
The answer is "a + b multiplied, a + b whole squared"
a + b whole cube identity is derived. It equals a cube + 3 a b squared + 3 a squared b + b cube.
Which of the following is equivalent to a minus b whole cube?
1
2
3
The answer is "a - b multiplied, a - b whole squared"
a minus b whole cube identity is derived. It equals a cube minus 3a squared b + 3a b squared minus b cube.
Comparing the two identities the second can be derived from the first.
a cube + b cube identity is derived. It equals a + b, multiplied, a squared minus a b + b squared.
a cube + b cube identity is derived. It equals a minus b, multiplied, a squared + a b + b squared..
Comparing the two identities the second can be derived from the first.
algebraic identities of cube are derived using identities of squares and multiplication of polynomials.
Algebraic Identities of Cube are listed.
Which of the following helps to find (a+b)^4 ?
1
2
3
The answer is 'both the above'

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