10. \( a+b-c+8+m^{2}+71 m n-14 m^{2}-65 m n+m^{3}-m^{2}-115 m^{2}+6 m^{3} \) 12. \( x^{2} y-x^{3} y^{2}+x^{2} y-8 x^{4} y-x^{2} y-10+x^{3} y^{2}-7 x^{3} y^{2}-9+ \) 13. \( 5 a^{x+1}-3 b^{x+2}-8 c^{x+3}-5 a^{x+1}-50+4 b^{x+2}-65-b^{3} \) 14. \( a^{m+2}-x^{m+3}-5+8-3 a^{m+2}+5 x^{m+3}-6+a^{m+2}-5 x^{n} \) 15. \( 0.3 a+0.4 b+0.5 c-0.6 a-0.7 b-0.9 c+3 a-3 b- \) 16. \( \frac{1}{2} a+\frac{1}{3} b+2 a-3 b-\frac{3}{4} a-\frac{1}{6} b+\frac{3}{4}-\frac{1}{2} \) 17. \( \frac{3}{5} m^{2}-2 m n+\frac{1}{10} m^{2}-\frac{1}{3} m n+2 m n-2 m^{2} \) 18. \( -\frac{3}{4} a^{2}+\frac{1}{2} a b-\frac{5}{6} b^{2}+2 \frac{1}{3} a^{2}-\frac{3}{4} a b+\frac{1}{6} b^{2}-\frac{1}{3} b^{2}-2 a b \) 19. \( 0.4 x^{2} y+31+\frac{3}{8} x y^{2}-0.6 y^{3}-\frac{2}{5} x^{2} y-0.2 x y^{2}+\frac{1}{4} y^{3}-6 \) 20. \( \frac{3}{25} a^{m-1}-\frac{7}{50} b^{m-2}+\frac{3}{5} a^{m-1}-\frac{1}{25} b^{m-2}-0.2 a^{m-1}+\frac{1}{5} b^{m-2} \)
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Bonus Knowledge
Let's take a stroll through the fun world of polynomial simplification! Did you know that these expressions can often be found in various aspects of physics and engineering? For example, solving equations like these can help engineers determine optimal materials for building safe structures or even in computing trajectories for rockets. Who knew you'd be channeling your inner rocket scientist while simplifying polynomials? Now, when tackling polynomial expressions, it’s crucial to keep an eye out for like terms, where coefficients can be combined. A common mistake is overlooking negative signs—those pesky little buggers can flip your final answer! So, always do a double-check, maybe even read the expression out loud to keep track of signs and terms. Happy simplifying!
