# How do you factor 4(x^3)-8(x^2)-25x+50?

Aug 29, 2016

$4 {x}^{3} - 8 {x}^{2} - 25 x + 50 = \left(2 x - 5\right) \left(2 x + 5\right) \left(x - 2\right)$

#### Explanation:

The difference of squares identity can be written:

${a}^{2} - {b}^{2} = \left(a - b\right) \left(a + b\right)$

We use this with $a = 2 x$ and $b = 5$ later...

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Note that the ratio of the first and second terms is the same as that between the third and fourth terms. So this cubic will factor by grouping:

$4 {x}^{3} - 8 {x}^{2} - 25 x + 50$

$= \left(4 {x}^{3} - 8 {x}^{2}\right) - \left(25 x - 50\right)$

$= 4 {x}^{2} \left(x - 2\right) - 25 \left(x - 2\right)$

$= \left(4 {x}^{2} - 25\right) \left(x - 2\right)$

$= \left({\left(2 x\right)}^{2} - {5}^{2}\right) \left(x - 2\right)$

$= \left(2 x - 5\right) \left(2 x + 5\right) \left(x - 2\right)$