x³ + 5 x² -4x -20 synthetic
Answer (1)
To solve x^3 + 5x^2 - 4x - 20 using synthetic division, we need to find a root of the polynomial.By the rational root theorem, possible roots are \pm 1, \pm 2, \pm 4, \pm 5, \pm 10, \pm 20.Let's test x = -5:\begin{array}{c|cccc}-5 & 1 & 5 & -4 & -20 \& & -5 & 0 & 20 \\hline& 1 & 0 & -4 & 0\end{array}Since the remainder is 0, x = -5 is a root, and (x+5) is a factor. The quotient is x^2 - 4.So, x^3 + 5x^2 - 4x - 20 = (x+5)(x^2 - 4)Now, we can factor x^2 - 4 as a difference of squares:x^2 - 4 = (x-2)(x+2)Therefore, x^3 + 5x^2 - 4x - 20 = (x+5)(x-2)(x+2)The roots are x = -5, x = 2, x = -2.
