Question

question
type the correct answer in each box.
consider the difference of cubes identity.
$a^3 - b^3 = (a - b)(a^2 + ab + b^2)$.
for the polynomial $x^3 - 64$,
$a = \boxed{}$ and $b = \boxed{}$.

Explanation:

Step1: Identify \(a^3\) in \(x^3 - 64\)

In the polynomial \(x^3 - 64\), we can see that \(x^3\) is a perfect cube. Comparing with \(a^3 - b^3\), we have \(a^3 = x^3\). To find \(a\), we take the cube root of \(x^3\). The cube root of \(x^3\) is \(x\) because \((x)^3 = x^3\). So, \(a = x\).

Step2: Identify \(b^3\) in \(x^3 - 64\)

In the polynomial \(x^3 - 64\), the constant term is \(-64\). Comparing with \(a^3 - b^3\), we have \(b^3 = 64\) (since the expression is \(x^3 - 64 = x^3 - 4^3\)). To find \(b\), we take the cube root of \(64\). The cube root of \(64\) is \(4\) because \(4^3 = 64\). So, \(b = 4\).

Answer:

\(a = \boldsymbol{x}\) and \(b = \boldsymbol{4}\)