Question

use the given information about the polynomial graph to write the equation. degree: 5, zeroes of multiplicity 2 at ( x = 3 ) and ( x = 1 ), and a zero of multiplicity 1 at ( x = -3 ), ( y )-intercept at ( (0,9) ). select one: a. ( y = 9(x - 3)^2(x - 1)^2(x + 3) ). b. ( y = -\frac{1}{3}(x - 3)^2(x - 1)(x + 3) ). c. ( y = 3(x - 3)^2(x - 1)^2(x + 3) ). d. ( y = \frac{1}{3}(x - 3)^2(x - 1)^2(x + 3) ).

Explanation:

Step1: Write general polynomial form

For zeros $x=a$ (multiplicity $m$), $x=b$ (multiplicity $n$), $x=c$ (multiplicity $p$), the polynomial is $y = k(x-a)^m(x-b)^n(x-c)^p$. Substituting given zeros:
$y = k(x-3)^2(x-1)^2(x+3)$

Step2: Solve for $k$ using y-intercept

Substitute $x=0, y=9$ into the equation:
$9 = k(0-3)^2(0-1)^2(0+3)$
Calculate the constants:
$9 = k(9)(1)(3) = 27k$
Solve for $k$:
$k = \frac{9}{27} = \frac{1}{3}$

Step3: Substitute $k$ into polynomial

$y = \frac{1}{3}(x-3)^2(x-1)^2(x+3)$

Answer:

D. $y = \frac{1}{3}(x - 3)^2(x - 1)^2(x + 3)$