How would the expression ( x³ + 8 ) be rewritten using the sum of cubes?
a. ( (x²)(x² – 2x⁴) )
b. ( (x²)(x² – 2x – 4) )
c. ( (x – 2)(x² – 2x⁴) )
d. ( (x²)(x² + 2x – 4) )
The Correct Answer and Explanation is :
To rewrite the expression ( x^3 + 8 ) using the sum of cubes formula, we need to recognize that both terms in the expression are perfect cubes.
Step 1: Recognizing the Sum of Cubes Formula
The general sum of cubes formula is:
[
a^3 + b^3 = (a + b)(a^2 – ab + b^2)
]
In this case:
- ( x^3 ) is ( a^3 ) where ( a = x )
- ( 8 ) is ( b^3 ) where ( b = 2 ), since ( 2^3 = 8 )
So, we can apply the sum of cubes formula with ( a = x ) and ( b = 2 ).
Step 2: Applying the Formula
Using the sum of cubes formula:
[
x^3 + 8 = (x + 2)\left(x^2 – (x)(2) + 2^2\right)
]
This simplifies as:
[
x^3 + 8 = (x + 2)(x^2 – 2x + 4)
]
Step 3: Matching the Choices
From the provided options, none of the choices perfectly match the expression ( (x + 2)(x^2 – 2x + 4) ). However, based on what we derived using the sum of cubes, the correct form is the simplified expression above.
Correct Formulation:
The correct answer is none of the options exactly as written. The correct expression should be ( (x + 2)(x^2 – 2x + 4) ).
Explanation of the Process:
- The term ( x^3 ) is a cube of ( x ) and ( 8 ) is a cube of ( 2 ).
- Using the sum of cubes formula helps break down the expression into a product of two factors: ( (x + 2) ) and ( (x^2 – 2x + 4) ).