Question

find all rational roots of g(x).
g(x) = -x⁴ - x³ + 5x² + 3x - 2
write your answer as a list of simplified values separated by commas, if there is more than one value.

Explanation:

Step1: Apply Rational Root Theorem

Possible rational roots: $\pm1, \pm2$

Step2: Test $x=1$

$g(1) = -(1)^4 - (1)^3 + 5(1)^2 + 3(1) - 2 = -1 -1 +5 +3 -2 = 4
eq 0$

Step3: Test $x=-1$

$g(-1) = -(-1)^4 - (-1)^3 + 5(-1)^2 + 3(-1) - 2 = -1 +1 +5 -3 -2 = 0$

Step4: Factor out $(x+1)$

Use polynomial division/synthetic division:
$g(x) = (x+1)(-x^3 + 0x^2 + 5x - 2) = -(x+1)(x^3 - 5x + 2)$

Step5: Test $x=2$ for cubic

$2^3 -5(2)+2 = 8-10+2=0$

Step6: Factor cubic as $(x-2)(x^2+2x-1)$

$x^3-5x+2=(x-2)(x^2+2x-1)$

Step7: Identify rational roots

Only rational roots are $x=-1, x=2$ (quadratic has irrational roots)

Answer:

$-1, 2$