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Answer :
To factor the expression [tex]\(3x^2 + 27\)[/tex], we can follow these steps:
1. Identify the Greatest Common Factor (GCF):
Start by finding the greatest common factor of the terms in the expression. Here, both terms [tex]\(3x^2\)[/tex] and [tex]\(27\)[/tex] are divisible by [tex]\(3\)[/tex].
2. Factor Out the GCF:
Once the GCF is identified, you can factor it out from each term in the expression:
[tex]\[
3x^2 + 27 = 3(x^2 + 9)
\][/tex]
This represents the expression factored to the greatest extent possible using real numbers.
3. Check the Factored Form:
To verify the factored expression, we can distribute [tex]\(3\)[/tex] again. Multiply [tex]\(3\)[/tex] by each term inside the parentheses:
[tex]\[
3(x^2 + 9) = 3 \cdot x^2 + 3 \cdot 9 = 3x^2 + 27
\][/tex]
Since the expanded form matches the original expression, our factored form is correct.
Now, it’s important to mention that [tex]\(x^2 + 9\)[/tex] does not factor further using real numbers, but if you are also asked to account for complex numbers, then it could be factored as:
[tex]\[
x^2 + 9 = (x + 3i)(x - 3i)
\][/tex]
So, the expression using complex numbers is:
[tex]\[
3(x^2 + 9) = 3(x + 3i)(x - 3i)
\][/tex]
If you're asked to use radicals and the imaginary unit [tex]\(i\)[/tex], then this would be the complete factorization over the complex numbers. However, without considering complex numbers, [tex]\(3(x^2 + 9)\)[/tex] is as simplified as it can realistically be.
1. Identify the Greatest Common Factor (GCF):
Start by finding the greatest common factor of the terms in the expression. Here, both terms [tex]\(3x^2\)[/tex] and [tex]\(27\)[/tex] are divisible by [tex]\(3\)[/tex].
2. Factor Out the GCF:
Once the GCF is identified, you can factor it out from each term in the expression:
[tex]\[
3x^2 + 27 = 3(x^2 + 9)
\][/tex]
This represents the expression factored to the greatest extent possible using real numbers.
3. Check the Factored Form:
To verify the factored expression, we can distribute [tex]\(3\)[/tex] again. Multiply [tex]\(3\)[/tex] by each term inside the parentheses:
[tex]\[
3(x^2 + 9) = 3 \cdot x^2 + 3 \cdot 9 = 3x^2 + 27
\][/tex]
Since the expanded form matches the original expression, our factored form is correct.
Now, it’s important to mention that [tex]\(x^2 + 9\)[/tex] does not factor further using real numbers, but if you are also asked to account for complex numbers, then it could be factored as:
[tex]\[
x^2 + 9 = (x + 3i)(x - 3i)
\][/tex]
So, the expression using complex numbers is:
[tex]\[
3(x^2 + 9) = 3(x + 3i)(x - 3i)
\][/tex]
If you're asked to use radicals and the imaginary unit [tex]\(i\)[/tex], then this would be the complete factorization over the complex numbers. However, without considering complex numbers, [tex]\(3(x^2 + 9)\)[/tex] is as simplified as it can realistically be.
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