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Garrett Tanner
Mathematics
24 March, 12:28
How do I factor 125x^3+8
+4
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Eliza Griffin
24 March, 12:48
0
Step 1:
53x3 - 8
Step 2:
Trying to factor as a Difference of Cubes:
2.1 Factoring: 125x3-8
Theory : A difference of two perfect cubes, a3 - b3 can be factored into
(a-b) • (a2 + ab + b2)
Proof : (a-b) • (a2+ab+b2) =
a3+a2b+ab2-ba2-b2a-b3 =
a3 + (a2b-ba2) + (ab2-b2a) - b3 =
a3+0+0+b3 =
a3+b3
Check : 125 is the cube of 5
Check : 8 is the cube of 2
Check : x3 is the cube of x1
Factorization is:
(5x - 2) • (25x2 + 10x + 4)
Trying to factor by splitting the middle term
2.2 Factoring 25x2 + 10x + 4
The first term is, 25x2 its coefficient is 25.
The middle term is, + 10x its coefficient is 10.
The last term, "the constant", is + 4
Step-1 : Multiply the coefficient of the first term by the constant 25 • 4 = 100
Step-2 : Find two factors of 100 whose sum equals the coefficient of the middle term, which is 10.
-100 + - 1 = - 101
-50 + - 2 = - 52
-25 + - 4 = - 29
-20 + - 5 = - 25
-10 + - 10 = - 20
-5 + - 20 = - 25
For tidiness, printing of 12 lines which failed to find two such factors, was suppressed
Observation : No two such factors can be found!
Conclusion : Trinomial can not be factored
Final result:
(5x - 2) • (25x2 + 10x + 4)
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