Subtract the polynomials $(8 x^3-5 x+6)-(3 x^3+2 x-4)$. What is the result?
A. $5 x^3+3 x+10$
B. $5 x^3-3 x+2$
C. $5 x^3-7 x+10$
D. $5 x^3-7 x+2
Answer (1)
Distribute the negative sign: ( 8 x 3 − 5 x + 6 ) − ( 3 x 3 + 2 x − 4 ) = 8 x 3 − 5 x + 6 − 3 x 3 − 2 x + 4 .
Combine x 3 terms: 8 x 3 − 3 x 3 = 5 x 3 .
Combine x terms: − 5 x − 2 x = − 7 x .
Combine constant terms: 6 + 4 = 10 . The result is 5 x 3 − 7 x + 10 .
Explanation
Understanding the Problem We are asked to subtract the polynomial ( 3 x 3 + 2 x − 4 ) from the polynomial ( 8 x 3 − 5 x + 6 ) . This means we need to perform the operation ( 8 x 3 − 5 x + 6 ) − ( 3 x 3 + 2 x − 4 ) .
Distributing the Negative Sign To subtract the polynomials, we distribute the negative sign to each term of the second polynomial: − ( 3 x 3 + 2 x − 4 ) = − 3 x 3 − 2 x + 4 .
Combining Like Terms Now, we combine like terms from both polynomials: ( 8 x 3 − 5 x + 6 ) + ( − 3 x 3 − 2 x + 4 ) .
Combining x^3 Terms Combine the x 3 terms: 8 x 3 − 3 x 3 = 5 x 3 .
Combining x Terms Combine the x terms: − 5 x − 2 x = − 7 x .
Combining Constant Terms Combine the constant terms: 6 + 4 = 10 .
Final Result Write the resulting polynomial: 5 x 3 − 7 x + 10 .
Examples
Polynomial subtraction is a fundamental operation in algebra and has practical applications in various fields. For instance, in engineering, you might use polynomial subtraction to determine the difference in the performance of two systems modeled by polynomials. Imagine you have two different engine designs, and their efficiency is described by polynomials. Subtracting one polynomial from the other helps you find the range of conditions where one engine outperforms the other, allowing engineers to make informed decisions about which design to implement. This ensures optimal performance and resource allocation.
