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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.
mathsIntroduction to Algebra, Polynomials, and IdentitiesBasic Algebraic Identities

### Algebraic Identities of Cube

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

click on the content to continue..

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'

slide-show version coming soon