Factorize the expression and divide them as directed:
$(2 x^{3} - 12 x^{2} + 16 x) \div (x - 2) (x - 4)$

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Solution

Step 1: Factorize the expression
$2 x^{3} - 12 x^{2} + 16 x$
$= 2 x [x^{2} - 6 x + 8]$
$= 2 x [x^{2} - (4 + 2) x + 8]$
$= 2 x [x^{2} - 4 x - 2 x + 8]$
$= 2 x [x (x - 4) - 2 (x - 4)]$
$= (2 x) (x - 2) (x - 4)$

Step 2: Perform division of the given expression
$\frac{2 x^{3} - 12 x^{2} + 16 x}{(x - 2) (x - 4)} = \frac{2 x (x - 2) (x - 4)}{(x - 2) (x - 4)}$
$= 2 x$
$∴ (2 x^{3} - 12 x^{2} + 16 x) \div (x - 2) (x - 4) = 2 x$

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