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Answer :
To factor the sum of two cubes, you can use the formula for factoring a sum of cubes:
[tex]\[ a^3 + b^3 = (a + b)(a^2 - ab + b^2) \][/tex]
In the given expression, [tex]\(6x^3 + 48\)[/tex], the first step is to look for a common factor in both terms. Here, both terms can be divided by 6:
[tex]\[ 6(x^3 + 8) \][/tex]
Next, notice that [tex]\(x^3 + 8\)[/tex] is a sum of cubes because [tex]\(8\)[/tex] can be written as [tex]\(2^3\)[/tex]. So, it can be expressed as:
[tex]\[ x^3 + 2^3 \][/tex]
Now apply the sum of cubes formula:
1. Identify [tex]\(a = x\)[/tex] and [tex]\(b = 2\)[/tex].
2. Substitute into the formula:
[tex]\[ x^3 + 2^3 = (x + 2)(x^2 - 2x + 4) \][/tex]
Bringing back the factor of 6 we initially factored out, the expression becomes:
[tex]\[ 6(x + 2)(x^2 - 2x + 4) \][/tex]
Thus, the factored form of the expression [tex]\(6x^3 + 48\)[/tex] is:
[tex]\[ 6(x + 2)(x^2 - 2x + 4) \][/tex]
[tex]\[ a^3 + b^3 = (a + b)(a^2 - ab + b^2) \][/tex]
In the given expression, [tex]\(6x^3 + 48\)[/tex], the first step is to look for a common factor in both terms. Here, both terms can be divided by 6:
[tex]\[ 6(x^3 + 8) \][/tex]
Next, notice that [tex]\(x^3 + 8\)[/tex] is a sum of cubes because [tex]\(8\)[/tex] can be written as [tex]\(2^3\)[/tex]. So, it can be expressed as:
[tex]\[ x^3 + 2^3 \][/tex]
Now apply the sum of cubes formula:
1. Identify [tex]\(a = x\)[/tex] and [tex]\(b = 2\)[/tex].
2. Substitute into the formula:
[tex]\[ x^3 + 2^3 = (x + 2)(x^2 - 2x + 4) \][/tex]
Bringing back the factor of 6 we initially factored out, the expression becomes:
[tex]\[ 6(x + 2)(x^2 - 2x + 4) \][/tex]
Thus, the factored form of the expression [tex]\(6x^3 + 48\)[/tex] is:
[tex]\[ 6(x + 2)(x^2 - 2x + 4) \][/tex]
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