1. \( 2 x^{5}-8 x^{4}-10 x^{3}=0 \)

To solve the equation \( 2x^{5}-8x^{4}-10x^{3}=0 \), we first factor out the greatest common factor, which is \( 2x^{3} \): \[ 2x^{3}(x^{2}-4x-5)=0 \] Now, we can set each factor to zero. The first factor gives us \( 2x^{3}=0 \) leading to \( x=0 \). The second factor, \( x^{2}-4x-5=0 \), can be solved using the quadratic formula \( x=\frac{-b \pm \sqrt{b^{2}-4ac}}{2a} \). Substituting \( a=1 \), \( b=-4 \), and \( c=-5 \): \[ x=\frac{4 \pm \sqrt{(-4)^{2}-4(1)(-5)}}{2(1)}=\frac{4 \pm \sqrt{16+20}}{2}=\frac{4 \pm \sqrt{36}}{2}=\frac{4 \pm 6}{2} \] This gives us the solutions \( x=5 \) and \( x=-1 \). So the final solutions are \( x=0, x=5, x=-1 \). --- In the 17th century, when polynomial equations began to capture the attention of mathematicians, solutions like these became foundational to algebra. Famous mathematicians like René Descartes and Isaac Newton developed methods for solving higher degree polynomials, setting the stage for modern algebra. Today, understanding how to factor and solve polynomials is a vital skill in both theoretical and applied mathematics! If you're venturing further into polynomial roots, consider diving into synthetic division when handling higher-degree polynomials or using the Rational Root Theorem to test possible rational roots. Mistakes often involve skipping steps in factoring or overlooking simpler roots like \( x=0 \). Always double-check your factors!