Answer:Step 1 :
Equation at the end of step 1 :
((16 • (n6)) + (23•5n3)) + 25
Step 2 :
Equation at the end of step 2 :
(24n6 + (23•5n3)) + 25
Step 3 :
Trying to factor by splitting the middle term
3.1 Factoring 16n6+40n3+25
The first term is, 16n6 its coefficient is 16 .
The middle term is, +40n3 its coefficient is 40 .
The last term, "the constant", is +25
Step-1 : Multiply the coefficient of the first term by the constant 16 • 25 = 400
Step-2 : Find two factors of 400 whose sum equals the coefficient of the middle term, which is 40 .
-400 + -1 = -401
-200 + -2 = -202
-100 + -4 = -104
-80 + -5 = -85
-50 + -8 = -58
-40 + -10 = -50
-25 + -16 = -41
-20 + -20 = -40
-16 + -25 = -41
-10 + -40 = -50
-8 + -50 = -58
-5 + -80 = -85
-4 + -100 = -104
-2 + -200 = -202
-1 + -400 = -401
1 + 400 = 401
2 + 200 = 202
4 + 100 = 104
5 + 80 = 85
8 + 50 = 58
10 + 40 = 50
16 + 25 = 41
20 + 20 = 40 That's it
Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above, 20 and 20
16n6 + 20n3 + 20n3 + 25
Step-4 : Add up the first 2 terms, pulling out like factors :
4n3 • (4n3+5)
Add up the last 2 terms, pulling out common factors :
5 • (4n3+5)
Step-5 : Add up the four terms of step 4 :
(4n3+5) • (4n3+5)
Which is the desired factorization
Trying to factor as a Sum of Cubes :
3.2 Factoring: 4n3+5
Theory : A sum 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 : 4 is not a cube !!
Step-by-step explanation: I believe that is the answer because i got it right.
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