Use the polynomial identity x3−a3=(x−a)(x2+ax+a2)

Question

to rewrite x3−64
. (1 point)
Responses

x3−64=(x−4)(x2+4x+16)
x cubed minus 64 equals left parenthesis x minus 4 right parenthesis left parenthesis x squared plus 4 x plus 16 right parenthesis

x3−64=(x+4)(x2−4x+16)
x cubed minus 64 equals left parenthesis x plus 4 right parenthesis left parenthesis x squared minus 4 x plus 16 right parenthesis

x3−64=(x−4)(x2−4x+16)
x cubed minus 64 equals left parenthesis x minus 4 right parenthesis left parenthesis x squared minus 4 x plus 16 right parenthesis

x3−64=(x+4)(x2+4x+16)

Answer 1

To rewrite \( x^3 - 64 \) using the polynomial identity \( x^3 - a^3 = (x - a)(x^2 + ax + a^2) \), we first identify \( a \) in this case.

Here, \( a^3 = 64 \) implies \( a = \sqrt[3]{64} = 4 \).

Now, we can use the identity:

\[ x^3 - 64 = x^3 - 4^3 = (x - 4)(x^2 + 4x + 4^2) \]

Calculating \( 4^2 \):

\[ 4^2 = 16 \]

Therefore, we have:

\[ x^3 - 64 = (x - 4)(x^2 + 4x + 16) \]

Thus, the correct rewrite is:

\[ x^3 - 64 = (x - 4)(x^2 + 4x + 16) \]

The response is:

x³ - 64 = (x - 4)(x² + 4x + 16).