FACTORING CUBIC POLYNOMIALS USING IDENTITIES

Factor the following cubic polynomials. 

Example 1 :

8x³ + 125y³

Solution :

 8x³ + 125y³ = (2x)³ + (5y)³

a³ + b³ = (a + b)(a² - ab + b²)

= (2x + 5y)[(2x)² - (2x)(5y) + (5y)²]

= (2x + 5y)(4x² - 10xy + 25y²) 

Example 2 :

27x³ - 8y³

Solution :

27x³ - 8y³ = (3x)³ - (2y)³

 a³ - b³ = (a - b)(a² + ab + b²)

= (3x - 2y)[(3x)² + (3x)(2y) + (2y)²] 

  = (3x - 2y)(9x² + 6xy + 4y²) 

Example 3 :

a⁶ - 64

Solution :

a⁶ - 64 = (a²)³ - 4³

 a³ - b³ = (a - b)(a² + ab + b²)

= (a² - 4)[(a²)² + a²(4) + 4²]

= (a² - 4)[(a²)² + 4a² + 4²]

= (a² - 4)[(a²)² + 4² + 4a²]

a² + b² = (a + b)² - 2ab

= (a² - 4)[(a² + 4)² - 2(a²)(4) + 4a²]

= (a² - 4)[(a² + 4)² - 8a² + 4a²]

= (a² - 4)[(a² + 4)² - 4a²]

= (a² - 2²)[(a² + 4)² - (2a)²]

a² - b² = (a + b)(a - b)

= (a + 2)(a - 2)(a² + 4 + 2a)(a² + 4 - 2a)

= (a + 2)(a - 2)(a² + 2a + 4)(a² - 2a + 4)

Example 4 :

x³ + 8y³ + 6 xy - 1

Solution :

= x³ + 8y³ + 6xy - 1

= x³ + (2y)³ + (-1)³ - 3(x)(2y)(-1)

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

= (x + 2y - 1)(x² + 4y² + 1 - 2xy + 2y - x)

Example 5 :

l³ - 8m³ - 27n³ - 18 lmn

Solution :

= l³ - 8m³ - 27n³ - 18 lmn

= l³ + (-2m)³ + (-3n)³ - 3(1)(-2m)(-3n)

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

= (l - 2m - 3n)(l² + 4m² + 9n² + 2lm - 6mn + 3ln)

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