Solution
We are told to find the polynomial that has the following roots -4, -2, 2.
Explanation
- In order to solve this question in the easiest manner, we will make use of the following rules:
1. Sum of roots
2. Product of roots
3. Sum of Product of roots.
1. Sum of roots:
The sum of roots of a cubic equation is defined as follows:
[tex]\begin{gathered} \text{Given the cubic equation:} \\ y=ax^3+bx^2+cx+d \\ \text{If the roots of the equation are: }\alpha,\beta,\gamma \\ \text{then,} \\ \\ \alpha+\beta+\gamma=-\frac{b}{a} \end{gathered}[/tex]
2. Product of roots:
The product of roots of a cubic equation is defined as follows:
[tex]\alpha\times\beta\times\gamma=-\frac{d}{a}[/tex]
3. Sum of Product of roots:
[tex]\alpha\beta+\alpha\gamma+\beta\gamma=\frac{c}{a}[/tex]
- We have been given these roots to be -4, -2, and 2. Thus, we can apply the 3 formulas defined above to find the correct equation.
- This is done below:
[tex]\begin{gathered} Let\alpha=-4,\beta=-2,\text{ and }\gamma=2 \\ Also,\text{ let }a=1 \\ \\ -4+(-2)+2=-\frac{b}{a}=-\frac{b}{1} \\ -6+2=-b \\ \\ \therefore b=4 \\ \\ -4(-2)(2)=-\frac{d}{a}=-\frac{d}{1} \\ \therefore d=-16 \\ \\ -4(-2)+(-4)(2)+(-2)(2)=\frac{c}{a}=\frac{c}{1} \\ 8-8-4=c \\ \\ \therefore c=-4 \end{gathered}[/tex]
Now that we have all the coefficients, we can write out the equation as follows:
[tex]f(x)=x^3+4x^2-4x-16[/tex]
Final Answer
The answer is
[tex]f(x)=x^3+4x^2-4x-16\text{ (OPTION 4)}[/tex]
