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

Question

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

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

Answers ( )

    0
    2021-09-10T16:13:25+00:00

    Answer:

    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

    0
    2021-09-10T16:13:38+00:00

    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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