Question

question 3 of 5
enter the correct answer in the box.
what is the factored form of this expression?
$64x^3 - 125$
write all factors in standard form.

Explanation:

Step1: Identify the formula for difference of cubes

The expression \(64x^3 - 125\) is a difference of cubes. The formula for factoring a difference of cubes is \(a^3 - b^3=(a - b)(a^2+ab + b^2)\).

Step2: Determine \(a\) and \(b\)

For \(64x^3\), we can write it as \((4x)^3\) (since \(4^3 = 64\) and \((x)^3=x^3\)), so \(a = 4x\). For \(125\), we can write it as \(5^3\) (since \(5^3=125\)), so \(b = 5\).

Step3: Apply the difference of cubes formula

Substitute \(a = 4x\) and \(b = 5\) into the formula \(a^3 - b^3=(a - b)(a^2+ab + b^2)\).
First, calculate \(a - b\): \(4x-5\).
Then, calculate \(a^2+ab + b^2\):
\(a^2=(4x)^2 = 16x^2\), \(ab=(4x)(5)=20x\), \(b^2 = 5^2=25\). So \(a^2+ab + b^2=16x^2 + 20x+25\).
Putting it together, the factored form is \((4x - 5)(16x^2+20x + 25)\).

Answer:

\((4x - 5)(16x^2 + 20x + 25)\)