what polynomial identity should be used to prove that 133 = 125 + 8?

difference of cubes

difference of squares

square of binomial

sum of cubes

The answer is Sum of Cubes.

Since there is sign +, we expect it to be the sum. Among all choices, only the last one includes summation, and that is sum of cubes.

Let's check it out.
Sum of cubes can be expresses as:
a³ + b³ = (a + b)(a² - ab + b²)

Let's see what are cube roots of 125 and 8: 
∛125 = ∛5·5·5 = ∛5³ = 5     ⇒     a = 5
∛8 = ∛2·2·2 = ∛2³ = 2     ⇒     b = 2

Now, let's implement that in the equation of sum of cubes:
5³ + 2³ = (5 + 2)(5² - 5·2 + 2²)
125 + 8 = 7 · (25 - 10 + 4)
133 = 7 · 19
133 = 133

Therefore, sum of cubes is the right answer.