How do you factor $16 x^{3} - 250$ $16x^3 - 250$ ?

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How do you factor $16 x^{3} - 250$ $16x^3 - 250$ ?

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Answer 1 · 2015-09-27T10:01:41

Separate out the common scalar factor $2$ $2$ , then use the difference of cubes identity to find:

$16 x^{3} - 250 = 2 (2 x - 5) (4 x^{2} + 10 x + 25)$ $16x^3-250 = 2(2x-5)(4x^2+10x+25)$

Explanation:

Both $16$ $16$ and $250$ $250$ are divisible by $2$ $2$ so separate that out first.

$16 x^{3} - 250 = 2 (8 x^{3} - 125)$ $16x^3-250 = 2(8x^3-125)$

Now $8 = 2^{3}$ $8 = 2^3$ and $125 = 5^{3}$ $125 = 5^3$

So we find:

$2 (8 x^{3} - 125)$ $2(8x^3-125)$

$= 2 ({(2 x)}^{3} - 5^{3})$ $=2((2x)^3-5^3)$

$= 2 (2 x - 5) ({(2 x)}^{2} + (2 x) \cdot 5 + 5^{2})$ $=2(2x-5)((2x)^2+(2x)*5+5^2)$

$= 2 (2 x - 5) (4 x^{2} + 10 x + 25)$ $=2(2x-5)(4x^2+10x+25)$

...using the difference of cubes identity:

$a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})$ $a^3 - b^3 = (a-b)(a^2+ab+b^2)$

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