Answer: The zeros of the given equation are 2, -2 and -0.667. The y-intercept of the graph is f(0) = -8.
Explanation:
The given expression is,
[tex]f(x)=3x^3+2x^2-12x-8[/tex]
To find zeros put f(x)=0
[tex]3x^3+2x^2-12x-8=0[/tex]
This equation is satisfied by x=2, so (x-2) is the factor of the given equation. By synthetic division or long division we can find the remaining factor.
[tex](x-2)(3x^2+8x+4)=0[/tex]
Use Factoring method.
[tex](x-2)(3x^2+6x+2x+4)=0[/tex]
[tex](x-2)(3x(x+2)+2(x+2)=0[/tex]
[tex](x-2)(3x+2)(x+2)=0[/tex]
Equate each factor equal to zero by using zero product property.
So the zeros of the given equation are 2, -2 and -0.667.
To find the y-intercept of the graph, put x=0.
[tex]f(0)=3(0)^3+2(0)^2-12(0)-8[/tex]
[tex]f(0)=-8[/tex]
So the y-intercept of the graph is f(0) = -8.
Since the higher degree is 3 , which is odd and the coefficient of higher degree is positive, therefore the ead behavior is defined as,
[tex]f(x)\rightarrow \infty \text{ as }\rightarrow \infty\\f(x)\rightarrow -\infty \text{ as }\rightarrow -\infty[/tex]
It means as x decreases unboundedly then the function decreases unboundedly. similarly x increases unboundedly then the function increases unboundedly.
