Factorization Of Algebraic Expressions

Factorization Of Algebraic Expressions Of The Form a³ + b³ + c³, When a + b + c = 0

Example 1:   Factorize (x – y)³ + (y – z)³ + (z – x)³

Solution:    Let x – y = a, y– z = b and z – x = c, then,

a + b + c = x – y + y – z + z –x = 0

∴ a³ + b³ + c³ = 3abc

⇒ (x – y)³ + (y – z)³ + (z – x)³ = 3 (x–y)(y – z)(z–x)

Example 2:    Factorize  (a²–b²)³ + (b²–c²)³+ (c²–a²)³

Solution:    We have,

Example 3:    (text{Simplify }frac-)}^{3}}+-)}^{3}}+-)}^{3}}}++})

Solution:    We have,



Example 4:    Find the value of x³ – 8y³ – 36 xy – 216, when x = 2y + 6.

Solution:    We have,

FACTORIZATION OF x³ ± y³

In order to factorize the algebraic expression expressible as the sum or difference of two cubes, we sue the following identities.

(i) x³ + y³ = (x + y) (x² – xy+ y²)

(ii) x³ – y³ = (x – y) (x² + xy + y²)

Example 5:    Factorize 27x³ + 64y³

Solution:    27x³ + 64y³ 

= (3x + 4y) {(3x)² – (3x) (4y) + (4y)²},

= (3x + 4y) (9x² – 12 xy + 16y²)

Example 6:    Factorize a³ + 3a²b + 3ab² + b³ – 8

Solution:    Factorize a³ + 3a²b + 3ab² + b³ – 8

= (a + b)³ –2³

= {(a+b) – 2} {(a +b)² +(a +b).2+2²}

= (a + b– 2) (a² + 2ab + b² +2a + 2b + 4)

Example 7:    Factorize : a³ – 0.216

Solution:    a³ – 0.216

= a³ – (0.6)³

= (a –0.6) [a² + 0.6a +(0.6)²]

= (a–0.6) (a² + 0.6 a + 0.36)

Example 8:    Factorize:

(i) (x+ 1)³ – (x–1)³ (ii) 8(x + y)³ – 27 (x–y)³

Solution:    

Example 9:    Factorize:  (i) x⁶ – y⁶ (ii) x¹² – y¹²

Solution:  

Example 10:   Prove that:

(frac{0.87times 0.87times 0.87+0.13times 0.13times 0.13}{0.87times 0.87-0.87times 0.13+0.13times 0.13}=1)

Solution:    We have,

FACTORIZATION OF x³ + y³ + z³ – 3 xyz

(i) In order to factorize the algebraic expressions of the form x³ + y³ + z³ – 3 xyz

We use the following identity:

(i) x³ + y³ + z³ – 3 xyz = (x+y+z) (x² + y² + z²–xy – yz – zx)

(ii) If x + y + z = 0, then x³ + y³ + z³ = 3xyz

Example 11:   Factorize: 8x³ + 27y³+ z³ – 18 xyz

Solution:    We have,

Example 12:    Factorize :  (a+b)³ + (b+c)³ + (c+a)³ – 3(a+ b) (b+c) (c+a)

Solution:    We have,

Example 13:    Resolve a³ – b³ + 1 + 3ab into factors

Solution:   a³ – b³ + 1 + 3ab

Example 14:    Factorize : 2√2 a³+ 8b³ – 27c³ + 18√2 abc

Solution:    2√2 a³+ 8b³ – 27c³ + 18√2 abc

Example 15:    Prove that:

a³ + b³ + c³ – 3abc =  1/2 (a+b+c) {(a–b)² +  (b–c)² + (c–a)²}

Solution:    We have,