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Thought-Process to Discover Knowledge

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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.

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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

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The content is presented in small-focused learning units to enable you to

think,

figure-out, &

learn.

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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)`

*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*

trek

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

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'

*your progress details*

Progress

*About you*

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'