Final Answer:
The factored form of the polynomial is \[ (5x + 4)(25x^2 - 20x + 16) \]
Step-by-step explanation:
To factor the polynomial \( 125x^3 + 64 \), we can look at the sum of cubes formula. The sum of cubes formula is given by:
\[ a^3 + b^3 = (a + b)(a^2 - ab + b^2) \]
The polynomial \( 125x^3 + 64 \) can be written as a sum of cubes because \( 125 \) is the cube of \( 5 \) and \( 64 \) is the cube of \( 4 \). So we have:
\[ 125x^3 + 64 = (5x)^3 + 4^3 \]
Now we can recognize \( a = 5x \) and \( b = 4 \), and apply the formula:
\[ a^3 + b^3 = (a + b)(a^2 - ab + b^2) \]
Substitute \( a = 5x \) and \( b = 4 \):
\[ (5x)^3 + 4^3 = (5x + 4)((5x)^2 - (5x)(4) + 4^2) \]
Now let's expand and simplify the terms:
\[ = (5x + 4)(25x^2 - 20x + 16) \]
Therefore, the factored form of the given polynomial \( 125x^3 + 64 \) is:
\[ (5x + 4)(25x^2 - 20x + 16) \]