73x3 - 1000
STEP
2
:
Trying to factor as a Difference of Cubes:
2.1 Factoring: 343x3-1000
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 : 343 is the cube of 7
Check : 1000 is the cube of 10
Check : x3 is the cube of x1
Factorization is :
(7x - 10) • (49x2 + 70x + 100)
Trying to factor by splitting the middle term
2.2 Factoring 49x2 + 70x + 100
The first term is, 49x2 its coefficient is 49 .
The middle term is, +70x its coefficient is 70 .
The last term, "the constant", is +100
Step-1 : Multiply the coefficient of the first term by the constant 49 • 100 = 4900
Step-2 : Find two factors of 4900 whose sum equals the coefficient of the middle term, which is 70 .
-4900 + -1 = -4901
-2450 + -2 = -2452
-1225 + -4 = -1229
-980 + -5 = -985
-700 + -7 = -707
-490 + -10 = -500
For tidiness, printing of 48 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 :
(7x - 10) • (49x2 + 70x + 100)
