The polynomial x3 + 8 is equal to
The Correct Answer and Explanation is:
The polynomial ( x^3 + 8 ) can be factored using the sum of cubes formula. The sum of cubes formula is:
[
a^3 + b^3 = (a + b)(a^2 – ab + b^2)
]
In this case, ( x^3 + 8 ) is a sum of cubes, where ( x^3 ) is the cube of ( x ), and ( 8 ) is the cube of ( 2 ) (since ( 2^3 = 8 )). Therefore, we can rewrite the polynomial as:
[
x^3 + 8 = x^3 + 2^3
]
Now, applying the sum of cubes formula, we substitute ( a = x ) and ( b = 2 ) into the formula:
[
x^3 + 2^3 = (x + 2)(x^2 – x \cdot 2 + 2^2)
]
Simplifying the terms inside the parentheses:
[
x^3 + 8 = (x + 2)(x^2 – 2x + 4)
]
Thus, the factored form of ( x^3 + 8 ) is:
[
x^3 + 8 = (x + 2)(x^2 – 2x + 4)
]
Explanation:
The polynomial ( x^3 + 8 ) is a sum of cubes because both ( x^3 ) and ( 8 ) are perfect cubes (with ( 8 = 2^3 )). The sum of cubes formula is a standard algebraic identity that allows us to factor such expressions efficiently. By recognizing that the given expression fits this identity, we apply the formula to break it into two factors: ( (x + 2) ), which is the sum of the cube roots, and ( (x^2 – 2x + 4) ), which is a quadratic factor that results from the formula. Factoring helps simplify expressions and makes it easier to solve polynomial equations or analyze their properties, such as finding roots or analyzing behavior.