Answer:
[tex]f(x)=2x^{3}+24x^{2}+82x+60[/tex]
Step-by-step explanation:
we know that
The roots of the polynomial are the values of x when the value of the polynomial f(x) is equal to zero
The roots of the polynomial function are
x=-6 -----> (x+6)=0
x=-5 -----> (x+5)=0
x=-1 -----> (x+1)=0
The equation of the cubic polynomial is
[tex]f(x)=a(x+6)(x+5)(x+1)[/tex]
where
a is the leading coefficient
Remember that
f(0)=60
That means ------> For x=0 the value of f(x) is equal to 60
substitute the value of x and the value of y in the function and solve for a
[tex]60=a(0+6)(0+5)(0+1)[/tex]
[tex]60=a(6)(5)(1)[/tex]
[tex]60=30a[/tex]
[tex]a=2[/tex]
so
[tex]f(x)=2(x+6)(x+5)(x+1)[/tex]
Applying the distributive property
Convert to expanded form
[tex]f(x)=2(x+6)(x+5)(x+1)\\\\f(x)=2(x+6)(x^{2}+x+5x+5)\\\\f(x)=2(x+6)(x^{2}+6x+5)\\\\f(x)=2(x^{3}+6x^{2}+5x+6x^{2}+36x+30)\\\\f(x)=2x^{3}+24x^{2}+82x+60[/tex]
