Question

FACTORIZE
${x}^{3} + 27$
$1 + 64 {x}^{3}$
$125 {a}^{3}+\frac{1}{8}$
$16 {x}^{4} + 54x$



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Answer 1

Note :

• (a + b)² = a² + b² + 2ab

• (a - b)² = a² + b² - 2ab

• (a + b)(a - b) = a² - b²

• (a + b)³ = a³ + b³ + 3ab•(a + b)

• (a - b)³ = a³ - b³ - 3ab•(a - b)

• a³ + b³ = (a + b)(a² + b² - ab)

• a³ - b³ = (a - b)(a² + b² + ab)

Solution :

1). x³ + 27

=> x³ + 27

=> x³ + 3³

=> (x + 3)(x² + 3² - x•3)

=> (x + 3)(x² + 9 - 3x)

=> (x + 3)(x² - 3x + 9)

2). 1 + 64x³

=> 1 + 64x³

=> 1 + 4³x³

=> 1³ + (4x)³

=> (1 + 4x){1² + (4x)² - 1•4x}

=> (1 + 4x)(1 + 16x² - 4x)

=> (1 + 4x)(16x² - 4x + 1)

3). 125a³ + ⅛

=> 125a³ + ⅛

=> 5³a³ + ½³

=> (5a)³ + (½)³

=> (5a + ½){(5a)² + (½)² - 5a•½}

=> (5a + ½)(25a² + ¼ - 5a/2)

=> (5a + ½)(25a² - 5a/2 + ¼)

4). 16x⁴ + 54x

=> 16x⁴ + 54x

=> 2x(8x³ + 27)

=> 2x(2³x³ + 3³)

=> 2x{(2x)³ + 3³}

=> 2x(2x + 3){(2x)² + 3² - 2x•3}

=> 2x(2x + 3)(4x² + 9 - 6x)

=> 2x(2x + 3)(4x² - 6x + 9)