Question

handwritten step-by-step work legibly and boxed your answer. please show work to receive credit.

1. the polynomial below has a zero at 6, use synthetic division to determine the remaining factors and graph the polynomial. please show all the zeros and box your answer. (referenced from section 6.6/ 3 points)

$g(x) = x^3 - 8x^2 - 3x + 90$

Explanation:

Step1: Set up synthetic division

Use root $6$ for $g(x)=x^3-8x^2-3x+90$:

$$\begin{array}{r|rrrr} 6 & 1 & -8 & -3 & 90 \\ \hline & & 6 & -12 & -90 \\ \hline & 1 & -2 & -15 & 0 \end{array}$$

Step2: Get quadratic factor

From the result, the quadratic is $x^2-2x-15$.

Step3: Factor the quadratic

Factor the quadratic expression:
$x^2-2x-15=(x-5)(x+3)$

Step4: Identify all zeros

Set each factor equal to 0:
$x-6=0 \implies x=6$; $x-5=0 \implies x=5$; $x+3=0 \implies x=-3$

Answer:

Remaining factors: $(x-5)(x+3)$
All zeros: $x=-3$, $x=5$, $x=6$
Fully factored polynomial: $g(x)=(x+3)(x-5)(x-6)$

Graph Notes:

• The cubic polynomial has a positive leading coefficient, so it falls to the left and rises to the right.
• It crosses the x-axis at $(-3,0)$, $(5,0)$, and $(6,0)$.
• The y-intercept is at $(0,90)$.
• The graph has a local maximum between $x=-3$ and $x=5$, and a local minimum between $x=5$ and $x=6$.