Simplify $(8 j^3-5 j^2-5)-(6 j^3-12 j^2+8 j-7)$
Answer (2)
To simplify the expression ( 8 j 3 − 5 j 2 − 5 ) − ( 6 j 3 − 12 j 2 + 8 j − 7 ) , distribute the negative sign, combine like terms, and the final result is 2 j 3 + 7 j 2 − 8 j + 2 .
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Distribute the negative sign: ( 8 j 3 − 5 j 2 − 5 ) − ( 6 j 3 − 12 j 2 + 8 j − 7 ) = 8 j 3 − 5 j 2 − 5 − 6 j 3 + 12 j 2 − 8 j + 7 .
Combine the j 3 terms: 8 j 3 − 6 j 3 = 2 j 3 .
Combine the j 2 terms: − 5 j 2 + 12 j 2 = 7 j 2 .
Combine the constant terms: − 5 + 7 = 2 .
The simplified expression is 2 j 3 + 7 j 2 − 8 j + 2 .
Explanation
Understanding the Problem We are given the expression ( 8 j 3 − 5 j 2 − 5 ) − ( 6 j 3 − 12 j 2 + 8 j − 7 ) and we want to simplify it. This involves distributing the negative sign and combining like terms.
Distributing the Negative Sign First, distribute the negative sign to each term in the second polynomial: ( 8 j 3 − 5 j 2 − 5 ) − ( 6 j 3 − 12 j 2 + 8 j − 7 ) = 8 j 3 − 5 j 2 − 5 − 6 j 3 + 12 j 2 − 8 j + 7
Combining Like Terms Next, we combine like terms. Group the terms with the same power of j together: ( 8 j 3 − 6 j 3 ) + ( − 5 j 2 + 12 j 2 ) + ( − 8 j ) + ( − 5 + 7 ) Now, perform the operations: 2 j 3 + 7 j 2 − 8 j + 2
Final Result Therefore, the simplified expression is 2 j 3 + 7 j 2 − 8 j + 2 .
Examples
Polynomials are used to model curves and shapes in engineering and computer graphics. Simplifying polynomial expressions like this one can help engineers optimize designs, or help game developers create realistic graphics more efficiently. For example, if j represents a parameter in a design, simplifying the polynomial helps understand how changes in j affect the overall outcome.
