Alright, let's explore these two algebraic expressions step-by-step.
### Expression 1: [tex]\(3q^3 + 15a^2\)[/tex]
We have the expression [tex]\(3q^3 + 15a^2\)[/tex].
To see if we can simplify or factor this expression, we notice that both terms have a numerical coefficient that's a multiple of 3. Thus, we can factor out the greatest common factor, which is 3:
[tex]\[ 3q^3 + 15a^2 = 3(q^3 + 5a^2) \][/tex]
So, the expression simplifies to:
[tex]\[ 3(q^3 + 5a^2) \][/tex]
This is as simplified as we can make it, as we cannot factor [tex]\(q^3 + 5a^2\)[/tex] any further over the set of real numbers.
### Expression 2: [tex]\(3a^3 + 3q^3 - 60q\)[/tex]
Next, we have the expression [tex]\(3a^3 + 3q^3 - 60q\)[/tex].
Similarly, we see that we can factor out the greatest common factor, which again is 3:
[tex]\[ 3a^3 + 3q^3 - 60q = 3(a^3 + q^3 - 20q) \][/tex]
Now we have:
[tex]\[ 3(a^3 + q^3 - 20q) \][/tex]
To explore [tex]\( a^3 + q^3 - 20q \)[/tex] further, we recognize that [tex]\( a^3 + q^3 \)[/tex] is a sum of cubes, but since there is an additional [tex]\(-20q\)[/tex] term, it complicates the usual factorization.
The sum of cubes can be factored as follows:
[tex]\[ a^3 + b^3 = (a + b)(a^2 - ab + b^2) \][/tex]
However, there isn't a straightforward way to factor [tex]\( a^3 + q^3 - 20q \)[/tex] beyond recognizing the common factor.
Thus, the simplified form of the second expression remains:
[tex]\[ 3(a^3 + q^3 - 20q) \][/tex]
### Summary
Thus, both expressions have been simplified as follows:
1. [tex]\(3q^3 + 15a^2\)[/tex] simplifies to:
[tex]\[ 3(q^3 + 5a^2) \][/tex]
2. [tex]\(3a^3 + 3q^3 - 60q\)[/tex] simplifies to:
[tex]\[ 3(a^3 + q^3 - 20q) \][/tex]
These are the most simplified forms of the given expressions.
