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Given the expression:

[tex]\[ 3q^3 + 15a^2 ; 3a^3 + 3q^3 - 60q \][/tex]

Simplify or factor the terms if possible.


Sagot :

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.