Answer:
The required polynomial is:
[tex]f\left(x\right)=x^3+8x^2+19x+12[/tex]
Step-by-step explanation:
Given the zeros
-4, -1, -3
Zeros can be written as:
x = -4, x = -1, x = -3
Finding the factors
x = -4
x +4 = 0
x = -1
x + 1 = 0
x = -3
x + 3 = 0
Thus, the factor are:
(x+4) (x+1) (x+3)
The original function can be computed by multiplying the factors
[tex]\left(x+4\right)\left(x+1\right)\left(x+3\right)=0[/tex]
as
[tex]\left(x+4\right)\left(x+1\right)=x^2+5x+4[/tex]
so the expression becomes
[tex]\left(x^2+5x+4\right)\left(x+3\right)=0[/tex]
Distribute parentheses
[tex]x^2x+x^2\cdot \:3+5xx+5x\cdot \:3+4x+4\cdot \:3=0[/tex]
[tex]x^2x+3x^2+5xx+5\cdot \:\:3x+4x+4\cdot \:\:3=0[/tex]
[tex]x^3+3x^2+5x^2+15x+4x+12=0[/tex]
Add similar elements: [tex]3x^2+5x^2=8x^2[/tex]
[tex]x^3+8x^2+15x+4x+12 = 0[/tex]
Add similar elements: [tex]15x+4x=19x[/tex]
[tex]x^3+8x^2+19x+12=0[/tex]
Thus, the required polynomial is:
[tex]f\left(x\right)=x^3+8x^2+19x+12[/tex]
Note: your answer choices are a little ambiguous, but I have explained the entire procedure. Hopefully, it will clear your concept.
