Question

algebra 2 362403 - 6 fy - 1
benchmark test 2
due feb 5 - 11:45 pm late
find the zeros of the function ( f(x)=x^{3}-8x^{2}+9x + 18 ), and describe the behavior of the graph at each zero.
a. the graph crosses the x - axis at - 1, 3, and 6.
b. the graph touches the x - axis at - 1 and crosses the x - axis at - 3 and 6.
c. the graph crosses the x - axis at - 1, 3, and 9.
d. the graph touches the x - axis at - 3 and - 1 and crosses the x - axis at 9.
question 19 of 30

Explanation:

Step1: Test rational roots

Use Rational Root Theorem: possible roots are $\pm1, \pm2, \pm3, \pm6, \pm9, \pm18$.
Test $x=-1$: $f(-1)=(-1)^3-8(-1)^2+9(-1)+18=-1-8-9+18=0$. So $(x+1)$ is a factor.

Step2: Polynomial division

Divide $x^3-8x^2+9x+18$ by $(x+1)$:
$$\frac{x^3-8x^2+9x+18}{x+1}=x^2-9x+18$$

Step3: Factor quadratic

Factor $x^2-9x+18$:
$x^2-9x+18=(x-3)(x-6)$

Step4: Identify zeros & behavior

Full factorization: $f(x)=(x+1)(x-3)(x-6)$. All roots have multiplicity 1. When a root has odd multiplicity, the graph crosses the x-axis at that root. Zeros are $x=-1, 3, 6$.

Answer:

A. The graph crosses the x-axis at -1, 3, and 6.