Answer:
[tex] f(x)= (x+4) (x-3) (x-5)[/tex]
If we solve the first two terms we got:
[tex] f(x)= (x^2 -3x +4x -12) (x-5) = (x^2 +x -12) (x-5)[/tex]
And multiplying the last two terms we got:
[tex] f(x)= x^3 -5x^2 +x^2 -5x -12 x+60[/tex]
[tex] f(x) = x^3-4x^2 -17 x +60[/tex]
And then we satisfy all the conditions and our final answer would be:
[tex] f(x) = x^3-4x^2 -17 x +60[/tex]
Step-by-step explanation:
For this case we want to find a polynomial with a degree of 3 given by this general expression:
[tex] f(x) = ax^3 +bx^2 +cx +d[/tex]
With the condition of a =1, and with the following zeros
[tex] x_1 = -4, x_2 = 3, x_3 = 5[/tex]
And we satisfy that [tex]f(x_1) = f(x_2) = f(x_3) =0[/tex]
and we can find this polynomial like this:
[tex] f(x)= (x+4) (x-3) (x-5)[/tex]
If we solve the first two terms we got:
[tex] f(x)= (x^2 -3x +4x -12) (x-5) = (x^2 +x -12) (x-5)[/tex]
And multiplying the last two terms we got:
[tex] f(x)= x^3 -5x^2 +x^2 -5x -12 x+60[/tex]
[tex] f(x) = x^3-4x^2 -17 x +60[/tex]
And then we satisfy all the conditions and our final answer would be:
[tex] f(x) = x^3-4x^2 -17 x +60[/tex]
