Which expression represents the correct factorization of the following polynomial?
27x^3+64

A.(3x+4)(3x^2−12x+64)

B.3x+4(9x^2+12x−16)

C.(3x+4)(9x^2−12x+16)

D.(9x+16)(3x^2−12x+4)

I'm pretty sure that this one is the correct factorization of the polynomial above: c) (3x+4)(9x^2-12x+16)

(3x−4)⋅(9x  

2

+12x+16)

See steps

Step by Step Solution:

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STEP

1

:

Equation at the end of step 1

 33x3 -  64

STEP

2

:

Trying to factor as a Difference of Cubes

2.1      Factoring:  27x3-64  

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 :  27  is the cube of  3  

Check :  64  is the cube of   4  

Check :  x3 is the cube of   x1

Factorization is :

            (3x - 4)  •  (9x2 + 12x + 16)  

Trying to factor by splitting the middle term

2.2     Factoring  9x2 + 12x + 16  

The first term is,  9x2  its coefficient is  9 .

The middle term is,  +12x  its coefficient is  12 .

The last term, "the constant", is  +16  

Step-1 : Multiply the coefficient of the first term by the constant   9 • 16 = 144  

Step-2 : Find two factors of  144  whose sum equals the coefficient of the middle term, which is   12 .

     -144    +    -1    =    -145  

     -72    +    -2    =    -74  

     -48    +    -3    =    -51  

     -36    +    -4    =    -40  

     -24    +    -6    =    -30  

     -18    +    -8    =    -26  

For tidiness, printing of 24 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 :

 (3x - 4) • (9x2 + 12x + 16)

C

Step-by-step explanation:

When factorising the sum of cubes, we use the formula

a^3+b^3=(a+b)(a^2−ab+b^2).

In this case of 27x3+64,

27x^3=a^3

64=b^3

Find a:

27x^3=a^3

3^√27x^3=3^√a^3

3x=a

Find b:

64=b^3

3^√64=3√b^3

4=b

Substitute a=3x and b=4 into

 (a+b)(a^2−ab+b^2)

(3x+4)((3x)^2−(3x×4)+4^2)

= (3x+4)(9x^2−12x+16)

(3x+4)(9x^2−12x+16) is the factorised form of 27x^3+64

answer: 414,377,008

step-by-step explanation:

just multiply it