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How do you factor 125-x^3?

User Molfar
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2 Answers

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10 votes

First Image is the answer the second is the steps

Best of Luck

How do you factor 125-x^3?-example-1
How do you factor 125-x^3?-example-2
User Joel Peltonen
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19 votes
19 votes

Explanation:

We have


125 - x {}^(3)

First, 125 is a perfect cube because


5 * 5 * 5 = 125

and

x^3 is a perfect cube because


x * x * x = x {}^(3)

so we can use the difference of cubes identity


( {x}^(3) - {y}^(3) ) = (x - y)( {x}^(2) + xy + {y}^(2) )

Let say we have two perfect cubes:

64 because 8×8×8=64

and 27 because 3×3×3=27 and let subtract


64 - 27

we know that


64 - 27 = 37

but using the difference of cubes identity we should get the same thing.

Remeber cube root of 64 is 4 and cube root of 27 is 3 so we have


(4 - 3)( {4}^(2) + 4(3) + 3 {}^(2) )


1(16 + 12 + 9) = 1(37) = 37

So the difference of cubes works for real numbers. This is a good way to help remeber the identity using real numbers.

Back on to the topic,

we know that 5 is cube root of 125 and x is the cube root of x^3 so we have


(5 - x)( {5}^(2) + 5x + {x}^(2) ) =


(5 - x)(25 + 5x + {x}^(2) )

User Woblob
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