Answer :
Certainly! Let's factor the expression [tex]\(18x^2 - 8\)[/tex] using two suitable methods: factoring out the Greatest Common Factor (GCF) and using the difference of squares rule.
### Method 1: Factoring out the GCF
1. Identify the GCF of the coefficients: The coefficients are 18 and 8. The GCF of 18 and 8 is 2.
2. Factor out the GCF:
[tex]\[ 18x^2 - 8 = 2 \cdot 9x^2 - 2 \cdot 4 \][/tex]
This can be rewritten by factoring out the GCF (which is 2):
[tex]\[ 18x^2 - 8 = 2(9x^2 - 4) \][/tex]
### Method 2: Using the Difference of Squares Rule
[tex]\[ 9x^2 - 4 \][/tex] can be recognized as a difference of squares.
1. Recall the difference of squares formula: [tex]\( a^2 - b^2 = (a + b)(a - b) \)[/tex].
2. Rewrite [tex]\(9x^2 - 4\)[/tex] as a difference of squares:
[tex]\[ 9x^2 = (3x)^2 \quad \text{and} \quad 4 = 2^2 \][/tex]
Thus,
[tex]\[ 9x^2 - 4 = (3x)^2 - 2^2 \][/tex]
3. Apply the difference of squares formula:
[tex]\[ (3x)^2 - 2^2 = (3x + 2)(3x - 2) \][/tex]
So putting it all together:
[tex]\[ 18x^2 - 8 = 2(9x^2 - 4) = 2(3x + 2)(3x - 2) \][/tex]
### Summary
The two best methods to factor the expression [tex]\(18x^2 - 8\)[/tex] are:
1. Factoring out the GCF: [tex]\(2(9x^2 - 4)\)[/tex]
2. Using the difference of squares rule: [tex]\(2(3x + 2)(3x - 2)\)[/tex]
Thus, the fully factored form of [tex]\(18x^2 - 8\)[/tex] is:
[tex]\[ 2(3x + 2)(3x - 2) \][/tex]
### Method 1: Factoring out the GCF
1. Identify the GCF of the coefficients: The coefficients are 18 and 8. The GCF of 18 and 8 is 2.
2. Factor out the GCF:
[tex]\[ 18x^2 - 8 = 2 \cdot 9x^2 - 2 \cdot 4 \][/tex]
This can be rewritten by factoring out the GCF (which is 2):
[tex]\[ 18x^2 - 8 = 2(9x^2 - 4) \][/tex]
### Method 2: Using the Difference of Squares Rule
[tex]\[ 9x^2 - 4 \][/tex] can be recognized as a difference of squares.
1. Recall the difference of squares formula: [tex]\( a^2 - b^2 = (a + b)(a - b) \)[/tex].
2. Rewrite [tex]\(9x^2 - 4\)[/tex] as a difference of squares:
[tex]\[ 9x^2 = (3x)^2 \quad \text{and} \quad 4 = 2^2 \][/tex]
Thus,
[tex]\[ 9x^2 - 4 = (3x)^2 - 2^2 \][/tex]
3. Apply the difference of squares formula:
[tex]\[ (3x)^2 - 2^2 = (3x + 2)(3x - 2) \][/tex]
So putting it all together:
[tex]\[ 18x^2 - 8 = 2(9x^2 - 4) = 2(3x + 2)(3x - 2) \][/tex]
### Summary
The two best methods to factor the expression [tex]\(18x^2 - 8\)[/tex] are:
1. Factoring out the GCF: [tex]\(2(9x^2 - 4)\)[/tex]
2. Using the difference of squares rule: [tex]\(2(3x + 2)(3x - 2)\)[/tex]
Thus, the fully factored form of [tex]\(18x^2 - 8\)[/tex] is:
[tex]\[ 2(3x + 2)(3x - 2) \][/tex]
