Answer :
To factor the expression [tex]\(48x^2 + 27\)[/tex] completely, let's follow these steps:
1. Find the greatest common factor (GCF):
- Look at the coefficients of the terms: 48 and 27.
- The greatest common factor of 48 and 27 is 3.
- Factor out 3 from the entire expression.
2. Factor out the GCF:
- Divide each term in the expression by the GCF, 3.
- [tex]\(48x^2 + 27\)[/tex] becomes [tex]\(3(16x^2 + 9)\)[/tex].
3. Factor the remaining expression:
- Look at the expression inside the parentheses: [tex]\(16x^2 + 9\)[/tex].
- Notice this doesn't factor further because it is in the form of a sum of squares: [tex]\((4x)^2 + 3^2\)[/tex].
- The sum of squares cannot be factored further over the set of real numbers.
So, the expression [tex]\(48x^2 + 27\)[/tex] is factored completely as [tex]\(3(16x^2 + 9)\)[/tex].
1. Find the greatest common factor (GCF):
- Look at the coefficients of the terms: 48 and 27.
- The greatest common factor of 48 and 27 is 3.
- Factor out 3 from the entire expression.
2. Factor out the GCF:
- Divide each term in the expression by the GCF, 3.
- [tex]\(48x^2 + 27\)[/tex] becomes [tex]\(3(16x^2 + 9)\)[/tex].
3. Factor the remaining expression:
- Look at the expression inside the parentheses: [tex]\(16x^2 + 9\)[/tex].
- Notice this doesn't factor further because it is in the form of a sum of squares: [tex]\((4x)^2 + 3^2\)[/tex].
- The sum of squares cannot be factored further over the set of real numbers.
So, the expression [tex]\(48x^2 + 27\)[/tex] is factored completely as [tex]\(3(16x^2 + 9)\)[/tex].
