Simplify: [tex] \left(2 x^3-x+5\right)-\left(6 x^3+x^2-x+1\right) [/tex]
(A) [tex]$8 x^3+x^2-2 x+6$[/tex]
(B) [tex]$-4 x^3-x^2+4$[/tex]
(C) [tex]$4 x^3-x^2-2 x-4$[/tex]
(D) [tex]$4 x^3-4$[/tex]
Answer (1)
Distribute the negative sign: 2 x 3 − x + 5 − 6 x 3 − x 2 + x − 1 .
Combine like terms: ( 2 x 3 − 6 x 3 ) + ( − x 2 ) + ( − x + x ) + ( 5 − 1 ) .
Simplify the expression: − 4 x 3 − x 2 + 4 .
The simplified expression is − 4 x 3 − x 2 + 4 .
Explanation
Understanding the Problem We are asked to simplify the expression ( 2 x 3 − x + 5 ) − ( 6 x 3 + x 2 − x + 1 ) . This involves combining like terms after distributing the negative sign.
Distributing the Negative Sign First, distribute the negative sign to each term in the second polynomial: 2 x 3 − x + 5 − 6 x 3 − x 2 + x − 1
Combining Like Terms Next, group the like terms together: ( 2 x 3 − 6 x 3 ) + ( − x 2 ) + ( − x + x ) + ( 5 − 1 ) Now, combine the like terms: − 4 x 3 − x 2 + 0 x + 4 So, the simplified expression is: − 4 x 3 − x 2 + 4
Final Answer Therefore, the simplified expression is − 4 x 3 − x 2 + 4 .
Examples
Polynomial simplification is a fundamental skill in algebra and is used in various real-world applications. For example, engineers use polynomials to model curves and surfaces in computer-aided design (CAD). Simplifying polynomial expressions allows them to optimize designs, reduce complexity, and improve efficiency in simulations and calculations. Understanding polynomial operations is also crucial in fields like physics, economics, and computer science, where mathematical models often involve polynomial functions.
