The polynomial $x^3+8$ is equal to
A. $(x+2)(x^2-2x+4)$
B. $(x-2)(x^2+2x+4)$
C. $(x+2)(x^2-2x+8)$
D. $(x-2)(x^2+2x+8)$
Answer (1)
Recognize x 3 + 8 as a sum of cubes.
Apply the sum of cubes formula: a 3 + b 3 = ( a + b ) ( a 2 − ab + b 2 ) .
Substitute a = x and b = 2 into the formula to get ( x + 2 ) ( x 2 − 2 x + 4 ) .
The correct factorization is ( x + 2 ) ( x 2 − 2 x + 4 ) .
Explanation
Recognizing the Sum of Cubes We are asked to factor the polynomial x 3 + 8 and choose the correct factorization from the given options. We can recognize that x 3 + 8 is a sum of cubes, which can be factored using the formula a 3 + b 3 = ( a + b ) ( a 2 − ab + b 2 ) .
Applying the Formula In this case, we have a = x and b = 2 . Applying the sum of cubes formula, we get:
x 3 + 8 = x 3 + 2 3 = ( x + 2 ) ( x 2 − ( x ) ( 2 ) + 2 2 ) = ( x + 2 ) ( x 2 − 2 x + 4 ) .
Identifying the Correct Option Comparing our result with the given options, we find that the correct factorization is ( x + 2 ) ( x 2 − 2 x + 4 ) .
Examples
Factoring polynomials like x 3 + 8 is useful in many areas of mathematics and engineering. For example, when designing a bridge, engineers use polynomials to model the forces acting on the structure. Factoring these polynomials can help identify critical points where the forces are maximized or minimized, ensuring the bridge's stability. Similarly, in signal processing, factoring polynomials is used to analyze and design filters that remove unwanted noise from signals. Understanding polynomial factorization allows engineers to predict and control the behavior of these systems, leading to more efficient and reliable designs.
