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Answer :
Sure! Let's go through the steps to factor the expression [tex]\(27x^2 - 48\)[/tex] completely:
1. Identify the Greatest Common Factor (GCF):
- First, we need to find the greatest common factor of the coefficients 27 and 48.
- The GCF of 27 and 48 is 3.
2. Factor out the GCF:
- We factor 3 out of each term in the expression [tex]\(27x^2 - 48\)[/tex].
- This gives us: [tex]\(3(9x^2 - 16)\)[/tex].
3. Recognize the Difference of Squares:
- Notice that [tex]\(9x^2 - 16\)[/tex] is a difference of squares. It can be written as [tex]\((3x)^2 - 4^2\)[/tex].
- The difference of squares formula is [tex]\(a^2 - b^2 = (a - b)(a + b)\)[/tex].
4. Apply the Difference of Squares Formula:
- Let [tex]\(a = 3x\)[/tex] and [tex]\(b = 4\)[/tex].
- So, [tex]\((3x)^2 - 4^2 = (3x - 4)(3x + 4)\)[/tex].
5. Write the Final Factored Expression:
- Combine all the parts together: [tex]\(3(3x - 4)(3x + 4)\)[/tex].
Thus, the completely factored form of the expression [tex]\(27x^2 - 48\)[/tex] is [tex]\(3(3x - 4)(3x + 4)\)[/tex].
1. Identify the Greatest Common Factor (GCF):
- First, we need to find the greatest common factor of the coefficients 27 and 48.
- The GCF of 27 and 48 is 3.
2. Factor out the GCF:
- We factor 3 out of each term in the expression [tex]\(27x^2 - 48\)[/tex].
- This gives us: [tex]\(3(9x^2 - 16)\)[/tex].
3. Recognize the Difference of Squares:
- Notice that [tex]\(9x^2 - 16\)[/tex] is a difference of squares. It can be written as [tex]\((3x)^2 - 4^2\)[/tex].
- The difference of squares formula is [tex]\(a^2 - b^2 = (a - b)(a + b)\)[/tex].
4. Apply the Difference of Squares Formula:
- Let [tex]\(a = 3x\)[/tex] and [tex]\(b = 4\)[/tex].
- So, [tex]\((3x)^2 - 4^2 = (3x - 4)(3x + 4)\)[/tex].
5. Write the Final Factored Expression:
- Combine all the parts together: [tex]\(3(3x - 4)(3x + 4)\)[/tex].
Thus, the completely factored form of the expression [tex]\(27x^2 - 48\)[/tex] is [tex]\(3(3x - 4)(3x + 4)\)[/tex].
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