Question

x⁴ - 5x³ + 11x² - 25x + 30 = 0
a. -2, -3, ±i√5
c. -2, 3, ±√5
b. 2, -3, ±√5
d. 2, 3, ±i√5
q find the zeros of the polynomial.

Explanation:

Step1: Test rational root candidates

Use Rational Root Theorem: possible roots are $\pm1,\pm2,\pm3,\pm5,\pm6,\pm10,\pm15,\pm30$.
Test $x=2$: $2^4 -5(2)^3 +11(2)^2 -25(2)+30=16-40+44-50+30=0$. So $x=2$ is a root.

Step2: Factor out $(x-2)$

Use polynomial division or synthetic division:
$$\frac{x^4 -5x^3 +11x^2 -25x+30}{x-2}=x^3 -3x^2 +5x -15$$

Step3: Factor the cubic polynomial

Group terms: $(x^3 -3x^2)+(5x-15)=x^2(x-3)+5(x-3)=(x-3)(x^2+5)$

Step4: Find remaining roots

Set $x^2+5=0$, solve for $x$:
$x^2=-5 \implies x=\pm i\sqrt{5}$

Step5: Compile all roots

The roots are $2, 3, \pm i\sqrt{5}$

Answer:

d. $2, 3, \pm i\sqrt{5}$