a) \( x^{4}+x-x^{3} y-y \) b) \( x^{3}-x-x^{2} y+y \) c) \( 6 x^{2}+x y-y^{2} \) d) \( a^{2}-b^{3}+2 b^{3} x^{2}-2 a^{2} x^{2} \) e) \( a^{2}+9 a+20 \) f) \( a^{2}-7 a+12 \) g) \( a^{2}-6 a+9 \) h) \( 6 x^{2}-x-2 \) i) \( 6 x^{2}+7 x y-3 y^{2} \) j) \( m^{4}+m^{2} n^{2}+n^{4} \) k) \( 15+14 x-8 x^{2} \) l) \( x^{6}+x^{3}-2 \) m) \( 2 \sqrt[3]{x^{2}}+5 \sqrt[3]{x}+2 \) n) \( 4 a^{2 n}-b^{2} \) n) \( x^{8}-y^{8} \) o) \( (\sqrt{6}-\sqrt{2}) \)
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Did you know that polynomials have been studied for centuries, with roots tracing back to ancient civilizations? The Babylonians were among the first to solve polynomial equations, laying the groundwork for algebra. Fast forward to today, where polynomial functions play crucial roles in fields like physics, economics, and engineering, helping to model everything from projectile motion to supply curves in economics. They’re the superheroes of mathematical modeling! If you’re working on factoring the polynomials listed, be on the lookout for common mistakes! One common error is not checking for a greatest common factor first before diving into more complex methods like synthetic division or the quadratic formula. Always double-check your work, especially signs when dealing with subtraction or negatives—one wrong sign can lead to a completely different polynomial!