Factorization Of Algebraic Expressions

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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,
Factorization-Of-Algebraic-Expressions-Example-1

Example 3:    \(\text{Simplify }\frac{{{({{a}^{2}}-{{b}^{2}})}^{3}}+{{({{b}^{2}}-{{c}^{2}})}^{3}}+{{({{c}^{2}}-{{a}^{2}})}^{3}}}{{{(a-b)}^{3}}+{{(b-c)}^{3}}+{{(c-a)}^{3}}}\)
Solution:    We have,
Factorization-Of-Algebraic-Expressions-Example-2
Example 4:    Find the value of x3 – 8y3 – 36 xy – 216, when x = 2y + 6.
Solution:    We have,
Factorization-Of-Algebraic-Expressions-Example-3

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:    
Factorization-Of-Algebraic-Expressions-Example-8

Example 9:    Factorize:  (i) x6 – y6 (ii) x12 – y12
Solution:  
Factorization-Of-Algebraic-Expressions-Example-9
Factorization-Of-Algebraic-Expressions-Example-9-1

Example 10:   Prove that:
\(\frac{0.87\times 0.87\times 0.87+0.13\times 0.13\times 0.13}{0.87\times 0.87-0.87\times 0.13+0.13\times 0.13}=1\)
Solution:    We have,
Factorization-Of-Algebraic-Expressions-Example-10

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,
Factorization-Of-Algebraic-Expressions-Example-11

Example 12:    Factorize :  (a+b)3 + (b+c)3 + (c+a)3 – 3(a+ b) (b+c) (c+a)
Solution:    We have,
Factorization-Of-Algebraic-Expressions-Example-12

Example 13:    Resolve a3 – b3 + 1 + 3ab into factors
Solution:   a3 – b3 + 1 + 3ab
Factorization-Of-Algebraic-Expressions-Example-13

Example 14:    Factorize : 2√2 a3+ 8b3 – 27c3 + 18√2 abc
Solution:    2√2 a3+ 8b3 – 27c3 + 18√2 abc
Factorization-Of-Algebraic-Expressions-Example-14

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,
Factorization-Of-Algebraic-Expressions-Example-15

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