\( \left. \begin{array} { | l | } \hline \text { Perform the indicated operation(s). } \\ { \left. \begin{array} { c } { ( x ^ { 2 } - x + 3 x ^ { 3 } - 7 ) + ( x ^ { 3 } - 5 x ^ { 2 } + 2 x ^ { 4 } - x + 6 ) } \\ { 2 x ^ { 4 } + x ^ { 3 } - 5 x ^ { 2 } - x + 6 } \\ { 2 x ^ { 4 } + x ^ { 3 } - 2 x ^ { 2 } - 3 x + 10 } \end{array} \right. } \\ \hline \left. \begin{array} { c } { ( 2 x ^ { 3 } - 5 x ^ { 2 } + 7 ) - ( x ^ { 3 } - 5 x ^ { 2 } + 2 x ^ { 4 } - x + 6 } \\ { ( 2 x ^ { 3 } + 5 x ^ { 2 } + x + 2 ) + ( 5 x ^ { 3 } + 7 x + 8 ) + ( x ^ { 2 } - x ^ { 3 } + 12 x ) - ( x ^ { 2 } - x ^ { 3 } + 12 x ) } \\ { 6 x ^ { 3 } + 4 x ^ { 2 } - 6 x + 6 } \end{array} \right. \end{array} \right. \)
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Bonus Knowledge
Let’s take a stroll down the polynomial lane! When adding polynomials, like the ones given, we collect like terms, which means we group the coefficients of the same degree together. So, for \( (x^2 - x + 3x^3 - 7) + (x^3 - 5x^2 + 2x^4 - x + 6) \), combine \(3x^3\) and \(x^3\) together, \(x^2\) and \(-5x^2\), and the constants like \(-7\) and \(6\). This gives us a new polynomial. Now, on the flip side, subtracting polynomials requires a keen eye! For \( (2x^3 - 5x^2 + 7) - (x^3 - 5x^2 + 2x^4 - x + 6) \), be careful to distribute that negative across all terms in the second polynomial. Once you do that, it's just a matter of combining like terms again—watch out for those pesky signs! 😊
