Answer:
[tex]y=-2x^3-4x^2+22x+24[/tex]Explanation:
The standard form of a cubic equation is given as:
[tex]y=ax^3+bx^2+cx+d[/tex]If the equation has x-intercepts of -4,-1, and 3, then we have:
At x=-4
[tex]\begin{gathered} At\text{ x=-4} \\ 0=a(-4)^3+b(-4)^2+c(-4)+d \\ -64a+16b-4c+d=0 \end{gathered}[/tex]At x=-1
[tex]\begin{gathered} 0=a\mleft(-1\mright)^3+b\mleft(-1\mright)^2+c\mleft(-1\mright)+d \\ -a+b-c+d=0 \end{gathered}[/tex]At x=3
[tex]\begin{gathered} 0=a\mleft(3\mright)^3+b\mleft(3\mright)^2+c\mleft(3\mright)+d \\ 27a+9b+3c+d=0 \end{gathered}[/tex]At the point (1,40)
[tex]\begin{gathered} 40=a\mleft(1\mright)^3+b\mleft(1\mright)^2+c\mleft(1\mright)+d \\ a+b+c+d=40 \end{gathered}[/tex]This gives us a system of equations with 4 unknowns.
[tex]\begin{gathered} -64a+16b-4c+d=0 \\ -a+b-c+d=0 \\ 27a+9b+3c+d=0 \\ a+b+c+d=40 \end{gathered}[/tex]If we solve this using a calculator, we have that:
[tex]a=-2,b=-4,c=22,d=24[/tex]Therefore, the cubic equation will be:
[tex]y=-2x^3-4x^2+22x+24[/tex]The graph of the equation is attached below:
We can clearly see the 4 given points on the graph above.
