Learn English – In math, round down or round up

The word for this is "factoring". "Factor" can be used as a verb, just like "distribute". When you do this:

3*a + 3*b

3(a + b)

you are factoring out the 3. (Present tense: "factor out the three".) When you do this:

x^2 + 5x + 6 = (x + 2)(x + 3)

you are factoring the polynomial.

"Factor" is also a noun. When you do this

100 = 2 * 2 * 5 * 5

you are finding the prime factors of 100.

"Distribute" is mainly used as a verb. When you do this:

5(a + b) = 5*a + 5*b

You are distributing the 5. But you can't call the 5s "distributors" or anything like that.

The property of multiplication and addition that allows you to do all this is called the distributive property.

You've already used the correct terminology: integer/integral part and fractional part.

The fractional part or decimal part of a non‐negative real number x {\displaystyle x} x is the excess beyond that number's integer part.

Fractional part - Wikipedia

You also might see noninteger part, as in:

Fractional part

The function frac(x) giving the fractional (noninteger) part of a real number x.

Weisstein, Eric W. "Fractional Part." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/FractionalPart.html