Learn the process of factorization of algebraic expressions of the form a^{3} + b^{3} + c^{3} – 3abc by going through the entire article. We have got you covered with everything like how to factorize expressions of the form a^{3} + b^{3} + c^{3} – 3abc by giving enough examples. Get to know the derivation and formula of the expression a^{3} + b^{3} + c^{3} – 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 a^{3} + b^{3} + c^{3} – 3abc make it easy for you to master the concept.

Identity a^{3} + b^{3} + c^{3} – 3abc = (a + b + c)(a^{2} + b^{2} + c^{2} – 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 a^{3} + b^{3} + c^{3} – 3abc Examples

**Example 1.
**Factorize u

^{3}+ v

^{3}– 3uv + 1?

**Solution:**

Given Expression = u

^{3}+ v

^{3}– 3uv + 1

= u

^{3}+ v

^{3}+ 1

^{3}– 3 ∙ u ∙ v ∙ 1

= (u + v + 1)(u

^{2}+ v

^{2}+ 1

^{2}– v ∙ 1 – 1 ∙ u – uv)

= (u + v + 1)(u

^{2}+v

^{2}– uv – u – v + 1)

**Example 2.
**Factorize 8a

^{3}+ 27b

^{3}– 90ab + 125?

**Solution:**

Given Expression = 8a

^{3}+ 27b

^{3}– 90ab + 125

= (2a)

^{3}+ (3b)

^{3}+ (5)

^{3}– 3 ∙ 2a ∙ 3b ∙ 5

= (2a + 3b + 5)(4x

^{2}+ 9y

^{2}– 6ab – 15b – 10a + 25)

**Example 3.
**Factorize the Expression 8x

^{3}+ y

^{3}+ 27z

^{3}–18xyz?

**Solution:**

Given Expression = 8x

^{3}+ y

^{3}+ 27z

^{3}–18xyz

=(2x)

^{3}+(y)

^{3}+(3z)

^{3}-3.2x.y.3z[As per Identity a

^{3}+ b

^{3}+ c

^{3}– 3abc = (a + b + c)(a

^{2}+ b

^{2}+ c

^{2}– bc – ca – ab)]

=(2x+y+3z)((2x)

^{2}+(y)

^{2}+(3z)

^{2}-y.3z-3z.2x-2x.y)

=(2x+y+3z)(4x

^{2}+y

^{2}+9z

^{2}-3yz-6xz-2xy)

**Example 4.
**Factorize the Expression 125x

^{3}+ 64y

^{3}+ 216z

^{3}– 360xyz?

**Solution:**

Given Expression = 125x

^{3}+ 64y

^{3}+ 216z

^{3}– 360xyz

=(5x)

^{3}+(4y)

^{3}+(6z)

^{3}-3.5x.4y.6z[As per Identity a

^{3}+ b

^{3}+ c

^{3}– 3abc = (a + b + c)(a

^{2}+ b

^{2}+ c

^{2}– bc – ca – ab)]

=(5x+4y+6z)((5x)

^{2}+(4y)

^{2}+(6z)

^{2}-4y.6z-6z.5x-5x.4y)

=(5x+4y+6z)(25x

^{2}+16y

^{2}+36z

^{2}-24yz-30xz-20xy)

**Example 5.
**Factorize the Expression 27a

^{3}+b

^{3}+8c

^{3}-18abc?

**Solution:**

Given Expression = 27a

^{3}+b

^{3}+8c

^{3}-18abc

= (3a)

^{3}+(b)

^{3}+(2c)

^{3}-3.3a.b.2c[As per Identity a

^{3}+ b

^{3}+ c

^{3}– 3abc = (a + b + c)(a

^{2}+ b

^{2}+ c

^{2}– bc – ca – ab)]

=(3a+b+2c)((3a)

^{2}+(b)

^{2}+(2c)

^{2}-b.2c-2c.3a-3a.b)

=(3a+b+2c)(9a

^{2}+b

^{2}+4c

^{2}-2bc-6ac-3ab)

**Example 6.
**Factorize the Expression 125u

^{3}+ v

^{3}–15uv + 1?

**Solution:**

Given Expression =125u

^{3}+ v

^{3}–15uv + 1

= (5u)

^{3}+(v)

^{3}+(1)

^{3}-3.5u.v.1[As per Identity a

^{3}+ b

^{3}+ c

^{3}– 3abc = (a + b + c)(a

^{2}+ b

^{2}+ c

^{2}– bc – ca – ab)]

=(5u+v+1)((5u)

^{2}+(v)

^{2}+(1)

^{2}-v.1-1.5u-5u.v)

=(5u+v+1)(25u

^{2}+v

^{2}+1-v-5u-5uv)