Factorization of Expressions of the Form a^3 + b^3 + c^3 – 3abc | How to Factor Expressions of Form a^3 + b^3 + c^3 – 3abc?

Learn the process of factorization of algebraic expressions of the form a³ + b³ + c³ – 3abc by going through the entire article. We have got you covered with everything like how to factorize expressions of the form a³ + b³ + c³ – 3abc by giving enough examples. Get to know the derivation and formula of the expression a³ + b³ + c³ – 3abc and use it while solving problems on the topic easily. Step by Step Explanations provided for all the questions on Factorization of Expressions of the Form a³ + b³ + c³ – 3abc make it easy for you to master the concept.

Identity a³ + b³ + c³ – 3abc =  (a + b + c)(a² + b² + c² – bc – ca – ab)

Read More:

• Factorization of Expressions of the Form a^3 + b^3
• Factorization of Expressions of the Form a^3 – b^3

Factoring Expressions of the Form a³ + b³ + c³ – 3abc Examples

Example 1.
Factorize  u³ + v³ – 3uv + 1?
Solution:
Given Expression = u³ + v³ – 3uv + 1
= u³ + v³ + 1³ – 3 ∙ u ∙ v ∙ 1
= (u + v + 1)(u² + v² + 1² – v ∙ 1 – 1 ∙ u – uv)
= (u + v + 1)(u² +v² – uv – u – v + 1)

Example 2.
Factorize 8a³ + 27b³ – 90ab + 125?
Solution:
Given Expression = 8a³ + 27b³ – 90ab + 125
= (2a)³ + (3b)³ + (5)³ – 3 ∙ 2a ∙ 3b ∙ 5
= (2a + 3b + 5)(4x² + 9y² – 6ab – 15b – 10a + 25)

Example 3.
Factorize the Expression 8x³ + y³+ 27z³ –18xyz?
Solution:
Given Expression = 8x³ + y³+ 27z³ –18xyz
=(2x)³ +(y)³ +(3z)³-3.2x.y.3z[As per Identity a³ + b³ + c³ – 3abc =  (a + b + c)(a² + b² + c² – bc – ca – ab)]
=(2x+y+3z)((2x)²+(y)²+(3z)²-y.3z-3z.2x-2x.y)
=(2x+y+3z)(4x²+y²+9z²-3yz-6xz-2xy)

Example 4.
Factorize the Expression 125x³ + 64y³ + 216z³– 360xyz?
Solution:
Given Expression = 125x³ + 64y³ + 216z³– 360xyz
=(5x)³+(4y)³+(6z)³-3.5x.4y.6z[As per Identity a³ + b³ + c³ – 3abc =  (a + b + c)(a² + b² + c² – bc – ca – ab)]
=(5x+4y+6z)((5x)²+(4y)²+(6z)²-4y.6z-6z.5x-5x.4y)
=(5x+4y+6z)(25x²+16y²+36z²-24yz-30xz-20xy)

Example 5.
Factorize the Expression 27a³+b³+8c³-18abc?
Solution:
Given Expression = 27a³+b³+8c³-18abc
= (3a)³+(b)³+(2c)³-3.3a.b.2c[As per Identity a³ + b³ + c³ – 3abc =  (a + b + c)(a² + b² + c² – bc – ca – ab)]
=(3a+b+2c)((3a)²+(b)²+(2c)²-b.2c-2c.3a-3a.b)
=(3a+b+2c)(9a²+b²+4c²-2bc-6ac-3ab)

Example 6.
Factorize the Expression 125u³ + v³ –15uv + 1?
Solution:
Given Expression =125u³ + v³ –15uv + 1
= (5u)³+(v)³+(1)³-3.5u.v.1[As per Identity a³ + b³ + c³ – 3abc =  (a + b + c)(a² + b² + c² – bc – ca – ab)]
=(5u+v+1)((5u)²+(v)²+(1)²-v.1-1.5u-5u.v)
=(5u+v+1)(25u²+v²+1-v-5u-5uv)