What is the quotient of $(x^3+8) \div(x+2)$?
A. $x^2+2 x+4$
B. $x^2-2 x+4$
C. $x^2+4$
D. $x^2-4$
Answer (1)
Recognize the expression as a sum of cubes.
Factor the sum of cubes: x 3 + 8 = ( x + 2 ) ( x 2 − 2 x + 4 ) .
Divide the factored form by ( x + 2 ) .
Cancel the common factor to get the quotient: x 2 − 2 x + 4 .
Explanation
Understanding the Problem We are asked to find the quotient of ( x 3 + 8 ) ÷ ( x + 2 ) . This is a polynomial division problem. We can solve this by factoring the numerator and then canceling common factors with the denominator.
Factoring the Sum of Cubes First, we recognize that x 3 + 8 is a sum of cubes. Specifically, x 3 + 8 = x 3 + 2 3 . The sum of cubes factorization formula is a 3 + b 3 = ( a + b ) ( a 2 − ab + b 2 ) . Applying this formula with a = x and b = 2 , we get: x 3 + 8 = x 3 + 2 3 = ( x + 2 ) ( x 2 − 2 x + 2 2 ) = ( x + 2 ) ( x 2 − 2 x + 4 )
Rewriting the Expression Now we can rewrite the original expression as: x + 2 x 3 + 8 = x + 2 ( x + 2 ) ( x 2 − 2 x + 4 )
Canceling Common Factors We can cancel the common factor of ( x + 2 ) from the numerator and the denominator: x + 2 ( x + 2 ) ( x 2 − 2 x + 4 ) = x 2 − 2 x + 4
Final Answer Therefore, the quotient of ( x 3 + 8 ) ÷ ( x + 2 ) is x 2 − 2 x + 4 .
Examples
Polynomial division is a fundamental concept in algebra and is used in various applications. For example, when designing a bridge, engineers use polynomial functions to model the load distribution and structural integrity. Dividing one polynomial by another helps them determine critical stress points and ensure the bridge's stability. Similarly, in computer graphics, polynomial division is used to create smooth curves and surfaces, essential for realistic rendering and animation. Understanding polynomial division allows for precise control and optimization in these complex systems.
