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Factorize each of the following expressions:$x^3 + 6x^2 + 12x + 16$
Given:
$x^3 + 6x^2 + 12x + 16$
To do:
We have to factorize the given expression.
Solution:
We know that,
$a^3 + b^3 = (a + b) (a^2 - ab + b^2)$
$a^3 - b^3 = (a - b) (a^2 + ab + b^2)$
Therefore,
$x^3 + 6x^2 + 12x + 16 = (x)^3 + 3(x^2)(2) + 3(x)(4) + (2)^3 + 8$ [Since $a^3 + 3a^2b + 3ab^2 +b^3 = (a + b)^3$]
$= (x + 2)^3 + 8$
$= (x + 2)^3 + (2)^3$
$= (x + 2 + 2) [(x + 2)^2 - (x + 2) \times 2 + (2)^2]$
$= (x + 4) (x^2 + 4x + 4 - 2x - 4 + 4)$
$= (x + 4) (x^2 + 2x + 4)$
Hence, $x^3 + 6x^2 + 12x + 16 = (x + 4) (x^2 + 2x + 4)$.
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