Answer :
To factor the sum of two cubes, we start with the expression [tex]\(6x^3 + 48\)[/tex].
### Step 1: Factor out the greatest common factor (GCF)
First, identify and factor out the greatest common factor from the terms in the expression. The GCF of [tex]\(6x^3\)[/tex] and [tex]\(48\)[/tex] is [tex]\(6\)[/tex]. So, we can factor out [tex]\(6\)[/tex] from the expression:
[tex]\[ 6(x^3 + 8) \][/tex]
### Step 2: Recognize the sum of cubes formula
The expression inside the parentheses, [tex]\(x^3 + 8\)[/tex], is a sum of cubes. The sum of cubes formula is:
[tex]\[ a^3 + b^3 = (a + b)(a^2 - ab + b^2) \][/tex]
In our case, [tex]\(x^3 + 8\)[/tex] can be rewritten using the sum of cubes formula where:
- [tex]\(a = x\)[/tex]
- [tex]\(b = 2\)[/tex] (since [tex]\(2^3 = 8\)[/tex])
### Step 3: Apply the sum of cubes formula
Substitute [tex]\(a = x\)[/tex] and [tex]\(b = 2\)[/tex] into the sum of cubes formula:
[tex]\[ x^3 + 8 = (x + 2)(x^2 - 2x + 4) \][/tex]
### Step 4: Rewrite the original expression
Now substitute this back into the expression after factoring out the GCF:
[tex]\[ 6(x^3 + 8) = 6(x + 2)(x^2 - 2x + 4) \][/tex]
### Conclusion
Therefore, the fully factored form of the given expression [tex]\(6x^3 + 48\)[/tex] is:
[tex]\[ 6(x + 2)(x^2 - 2x + 4) \][/tex]
This is the result of factoring the initial sum of cubes expression.
### Step 1: Factor out the greatest common factor (GCF)
First, identify and factor out the greatest common factor from the terms in the expression. The GCF of [tex]\(6x^3\)[/tex] and [tex]\(48\)[/tex] is [tex]\(6\)[/tex]. So, we can factor out [tex]\(6\)[/tex] from the expression:
[tex]\[ 6(x^3 + 8) \][/tex]
### Step 2: Recognize the sum of cubes formula
The expression inside the parentheses, [tex]\(x^3 + 8\)[/tex], is a sum of cubes. The sum of cubes formula is:
[tex]\[ a^3 + b^3 = (a + b)(a^2 - ab + b^2) \][/tex]
In our case, [tex]\(x^3 + 8\)[/tex] can be rewritten using the sum of cubes formula where:
- [tex]\(a = x\)[/tex]
- [tex]\(b = 2\)[/tex] (since [tex]\(2^3 = 8\)[/tex])
### Step 3: Apply the sum of cubes formula
Substitute [tex]\(a = x\)[/tex] and [tex]\(b = 2\)[/tex] into the sum of cubes formula:
[tex]\[ x^3 + 8 = (x + 2)(x^2 - 2x + 4) \][/tex]
### Step 4: Rewrite the original expression
Now substitute this back into the expression after factoring out the GCF:
[tex]\[ 6(x^3 + 8) = 6(x + 2)(x^2 - 2x + 4) \][/tex]
### Conclusion
Therefore, the fully factored form of the given expression [tex]\(6x^3 + 48\)[/tex] is:
[tex]\[ 6(x + 2)(x^2 - 2x + 4) \][/tex]
This is the result of factoring the initial sum of cubes expression.
