Question

select all zeros of the polynomial function. h(x)=x^4 - x^3 + 7x^2 - 9x - 18

Explanation:

Step1: Use Rational Root Theorem

By the Rational Root Theorem, possible rational zeros are $\pm1, \pm2, \pm3, \pm6, \pm9, \pm18$. Test $x=2$:
$h(2)=2^4 - 2^3 + 7(2)^2 - 9(2) - 18 = 16 - 8 + 28 - 18 - 18 = 0$.
Test $x=-1$:
$h(-1)=(-1)^4 - (-1)^3 + 7(-1)^2 - 9(-1) - 18 = 1 + 1 + 7 + 9 - 18 = 0$.

Step2: Factor out $(x-2)(x+1)$

First, expand $(x-2)(x+1)=x^2 - x - 2$. Use polynomial long division or synthetic division to divide $h(x)$ by $x^2 - x - 2$:
$\frac{x^4 - x^3 + 7x^2 - 9x - 18}{x^2 - x - 2} = x^2 + 9$.

Step3: Solve $x^2 + 9 = 0$

Rearrange the equation:
$x^2 = -9$
Take square roots:
$x = \pm 3i$

Answer:

The zeros of the polynomial are $2$, $-1$, $3i$, and $-3i$