Find a polynomial function with following characteristics.
Degree: 4
Zero: 4 , multiplicity: 2
Zero: $- 3$, multiplicity: 2
Falls to the left,
Falls to the right
Absolute value of the leading coefficient is one
A) $y = x ^ { 4 } + x ^ { 3 } - 23 x ^ { 2 } + 24 x - 12$
B) $y = - x ^ { 4 } + x ^ { 3 } - 48 x ^ { 2 } + 9$
C) $y = x ^ { 4 } - 6 x ^ { 3 } - 18 x ^ { 2 } + 25 x + 48$
D) $y = - x ^ { 4 } + 2 x ^ { 3 } + 23 x ^ { 2 } - 24 x - 144$
E) $y = - x ^ { 4 } + 2 x ^ { 3 } - 24 x - 36$

Question

Quizplus · Accepted Answer

Since the polynomial function has degree 4 and two zeros with multiplicity 2, we know that it can be factored in the following way:

$$f(x) = a(x-4)^2(x+3)^2$$

The falls to the left and falls to the right condition implies that the leading coefficient must be negative.

To satisfy the absolute value of the leading coefficient is one, let's choose $a = -1$.

Therefore,

$$f(x) = -(x-4)^2(x+3)^2$$

Expanding this expression gives us

$$f(x) = -x^4+2x^3+23x^2-24x-144 $$

which matches choice D.

Find a Polynomial Function with Following Characteristics −3- 3−3 , Multiplicity: 2 Falls to the Left, Falls to the 4

Find a Polynomial Function with Following Characteristics $- 3$ , Multiplicity: 2
Falls to the Left,
Falls to the 4