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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.continue

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



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

  • `(a+b)(a+b)^2`
  • `(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)(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 `a^3-a^3 = 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 `a^3-a^3 = 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`.

`a^3+c^3``=(a+c)(a^2-ac+c^2)`

substituting `c=-b`
`a^3-b^3``=(a-b)(a^2+ab+c^2)`

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.

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

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)`

Solved Exercise Problem:

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
  • both the above

The answer is 'both the above'

                            
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