solve f(x)=0 given the following function:

Question

f(x)=x⁵-4x⁴-5x³-8x²+32x+40

Answer 1

By Rational Zeros Theorem we have possible rational zeros: ±1,±2,±4,±5,±8,±10,±20,±40 (factors of 40).

Let try x = 2 using synthetic division, then x = 5, the x = -1

2 | 1 -4 -5 -8 32 40

2 -4 -18 -52 -40

5 | 1 -2 -9 -26 -20 | 0

5 15 30 20

-1 | 1 3 6 4 | 0

-1 -2 -4

1 2 4 | 0

Last quotient: x² + 2x + 4 = 0, x = -1 ± i√3'

Solution set of equation:{-1,2,5,-1±i√3}

Answer 2

Descartes Rule of Signs indicates 2 or 0 positive real root.

Therefore there are 3 or 1 negative roots. The other two would be imaginary.

Using synthetic substitution one zero is 2

The suppressed polynomial is x⁴ - 2x³ - 9x² - 26x - 20