Answer :
To factor the polynomial [tex]\( x^3 + 8 \)[/tex], we need to recognize it as a sum of cubes. The sum of cubes formula is:
[tex]\[ a^3 + b^3 = (a + b)(a^2 - ab + b^2) \][/tex]
In this problem, [tex]\( a = x \)[/tex] and [tex]\( b = 2 \)[/tex], because [tex]\( x^3 + 8 \)[/tex] can be rewritten as [tex]\( x^3 + 2^3 \)[/tex].
Using the formula:
- [tex]\( a = x \)[/tex]
- [tex]\( b = 2 \)[/tex]
Plug these values into the sum of cubes formula:
[tex]\[ x^3 + 2^3 = (x + 2)\left(x^2 - x \cdot 2 + 2^2\right) \][/tex]
Simplify inside the parentheses:
[tex]\[ x^3 + 8 = (x + 2)\left(x^2 - 2x + 4\right) \][/tex]
So, the factored form of the polynomial [tex]\( x^3 + 8 \)[/tex] is:
[tex]\[ (x + 2)\left(x^2 - 2x + 4\right) \][/tex]
Therefore, the correct answer is:
[tex]\[ (x+2)\left(x^2-2x+4\right) \][/tex]
[tex]\[ a^3 + b^3 = (a + b)(a^2 - ab + b^2) \][/tex]
In this problem, [tex]\( a = x \)[/tex] and [tex]\( b = 2 \)[/tex], because [tex]\( x^3 + 8 \)[/tex] can be rewritten as [tex]\( x^3 + 2^3 \)[/tex].
Using the formula:
- [tex]\( a = x \)[/tex]
- [tex]\( b = 2 \)[/tex]
Plug these values into the sum of cubes formula:
[tex]\[ x^3 + 2^3 = (x + 2)\left(x^2 - x \cdot 2 + 2^2\right) \][/tex]
Simplify inside the parentheses:
[tex]\[ x^3 + 8 = (x + 2)\left(x^2 - 2x + 4\right) \][/tex]
So, the factored form of the polynomial [tex]\( x^3 + 8 \)[/tex] is:
[tex]\[ (x + 2)\left(x^2 - 2x + 4\right) \][/tex]
Therefore, the correct answer is:
[tex]\[ (x+2)\left(x^2-2x+4\right) \][/tex]
