Question

select all zeros of the polynomial function.\\(g(x)=x^4 + 4x^3 + 7x^2 + 16x + 12\\)\\(-4\\) \\(-3\\) \\(-2\\) \\(-1\\) \\(0\\)\\(1\\) \\(2\\) \\(3\\) \\(4\\) \\(5\\)\\(-5i\\) \\(-4i\\) \\(-3i\\) \\(-2i\\) \\(-i\\)\\(i\\) \\(2i\\) \\(3i\\) \\(4i\\) \\(5i\\)

Explanation:

Step1: Use Rational Root Theorem

Test possible rational roots: $\pm1, \pm2, \pm3, \pm4, \pm6, \pm12$.
Test $x=-1$: $(-1)^4 + 4(-1)^3 +7(-1)^2 +16(-1)+12=1-4+7-16+12=0$.

Step2: Factor out $(x+1)$

Use polynomial division or synthetic division:
$$\frac{x^4+4x^3+7x^2+16x+12}{x+1}=x^3+3x^2+4x+12$$

Step3: Factor the cubic polynomial

Group terms: $(x^3+3x^2)+(4x+12)=x^2(x+3)+4(x+3)=(x^2+4)(x+3)$

Step4: Find all zeros

Set each factor to 0:

1. $x+1=0 \implies x=-1$
2. $x+3=0 \implies x=-3$
3. $x^2+4=0 \implies x^2=-4 \implies x=\pm2i$

Answer:

-3, -1, $2i$, $-2i$