Factor the polynomial
64x^3-27
64 = 4 ^ 3
64 x ^ 3 = ( 4 x ) ^ 3
27 = 3 ^ 3
64 x ^ 3 - 27 = ( 4 x ) ^ 3 - 3 ^ 3
a ^ 3 - b ^ 3 = ( a - b ) ( a ^ 2 + a b + b ^ 2 )
In this case:
64 x ^ 3 - 27 = ( 4 x ) ^ 3 - 3 ^ 3 = ( 4 x - 3 ) * [ ( 4 x ) ^ 2 + 4 x * 3 + 3 ^ 2 ] =
( 4 x - 3 ) * ( 16 x ^ 2 + 12 x + 9 )
To factor the polynomial 64x^3 - 27, we first notice that it is in the form of a difference of cubes:
a^3 - b^3 = (a - b)(a^2 + ab + b^2)
In this case, a = 4x and b = 3:
64x^3 - 27 = (4x)^3 - 3^3 = (4x - 3)((4x)^2 + (4x)(3) + 3^2)
Simplifying further:
(4x - 3)(16x^2 + 12x + 9)
Therefore, the factored form of the polynomial 64x^3 - 27 is (4x - 3)(16x^2 + 12x + 9).
To factor the polynomial 64x^3 - 27, we can use a special formula called the difference of cubes. This formula states that for any two numbers a and b, the difference of cubes can be factored as:
a^3 - b^3 = (a - b)(a^2 + ab + b^2)
In our case, we have 64x^3 - 27, which can be rewritten as (4x)^3 - 3^3. So, applying the difference of cubes formula:
64x^3 - 27 = (4x - 3)((4x)^2 + (4x)(3) + 3^2)
Simplifying further:
64x^3 - 27 = (4x - 3)(16x^2 + 12x + 9)
So, the factored form of the polynomial 64x^3 - 27 is (4x - 3)(16x^2 + 12x + 9).