## Factor completely. 16n^6+40n^3+25

Question

Factor completely.
16n^6+40n^3+25

in progress 0
2 weeks 2021-09-10T16:12:08+00:00 2 Answers 0

Step-by-step explanation:

(a + b)² = a² +2ab + b²

16n⁶ + 40n³ +25 = 4²*(n³)² + 2* 4n³ * 5   + 5²

=(4n³)² + 2 *4n³*5 + 5²

= (4n³ + 5)²

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.