Question

1. which of the following is a factor of $x^3 - 64$?

a) $x^2 + 4x + 16$ b) $x - 8$ c) $x + 4$ d) $x^2 - 4x + 16$
options: c, a, d, b

Explanation:

Step1: Recall the difference of cubes formula

The formula for the difference of cubes is \(a^3 - b^3=(a - b)(a^2+ab + b^2)\).

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

We can rewrite \(x^3 - 64\) as \(x^3-4^3\), so \(a = x\) and \(b = 4\).

Step3: Apply the difference of cubes formula

Using the formula \(a^3 - b^3=(a - b)(a^2+ab + b^2)\), we substitute \(a=x\) and \(b = 4\) to get:
\(x^3-4^3=(x - 4)(x^2+4x + 16)\)
So, one of the factors of \(x^3 - 64\) is \(x^2+4x + 16\) (option a).

Answer:

a) \(x^2+4x+16\)