Find a polynomial function with following characteristics. Degree: 4
Zero: $- 1$, multiplicity: 2
Zero: $- 3$, multiplicity: 2
Falls to the left,
Falls to the right
Absolute value of the leading coefficient is one
A) $y = x ^ { 4 } - 4 x ^ { 3 } + 22 x ^ { 2 } + 24 x + 3$
B) $y = - x ^ { 4 } - 4 x ^ { 3 } + 12 x ^ { 2 } + 9$
C) $y = x ^ { 4 } - 6 x ^ { 3 } - 18 x ^ { 2 } + 10 x + 3$
D) $y = - x ^ { 4 } - 8 x ^ { 3 } - 22 x ^ { 2 } - 24 x - 9$
E) $y = - x ^ { 4 } - 8 x ^ { 3 } - 24 x + 9$

Question

Quizplus · Accepted Answer

The given information tells us that the polynomial must have the factors $(x+1)^2$ and $(x+3)^2$. We can start building the polynomial by multiplying these factors together: 
\begin{align*}
(x + 1)^2 (x + 3)^2 &= (x^2 + 2x + 1)(x^2 + 6x + 9)\
&= x^4 + 8x^3 + 22x^2 + 24x + 9
\end{align*}
Option D matches this expression except for the sign of the leading coefficient. Since we want the absolute value of the leading coefficient to be 1, we can multiply the entire polynomial by $-1$ to get: 
$$y = -x^4 - 8x^3 - 22x^2 -24x - 9 $$ 
This polynomial satisfies all the given criteria: it is degree 4, has zeros of $-1$ and $-3$ each with multiplicity 2, falls to the left and right (due to the negative leading coefficient), and has an absolute value of the leading coefficient equal to 1.

Find a Polynomial Function with Following Characteristics −1- 1−1 , Multiplicity: 2 Zero

Find a Polynomial Function with Following Characteristics $- 1$ , Multiplicity: 2
Zero