To do so, we need to find the values of x for which f(x)=0.
f(x)=0 ⇔ 3x^4-6x^2+3=0
To solve the given equation, we will start by factoring out the greatest common factor.
To find solutions, we will need to define another variable. If we let z=x^2, we can rewrite the equation in terms of the z-variable.
x^4-2x^2+1=0 ⇔ z^2-2z+1=0
Note that the above equation in terms of z is a quadratic equation. Thus, we can solve it using the Quadratic Formula.
az^2+bz+c=0 ⇔ z=- b±sqrt(b^2-4ac)/2a
Let's identify a, b, and c.
z^2-2z+1=0 ⇔ 1z^2+( - 2)z+ 1=0
We see that a= 1, b= - 2, and c= 1. Let's substitute these values into the formula, and solve for z.
We found that the zeros of f(x)=3x^4-6x^2+3 are x=- 1 and x=1.
Graph
To draw the graph of the function, we will find some additional points and consider the end behavior. Let's use a table to find additional points.
x
3x^4-6x^2+3
f(x)=3x^4-6x^2+3
- 2
3( - 2)^4-6( - 2)^2+3
27
0
3( 0)^4-6( 0)^2+3
3
2
3( 2)^4-6( 2)^2+3
27
The points ( - 2, 27), ( 0, 3), and ( 2, 27) are on the graph of the function. We can also see that the leading coefficient is 3, which is a positive number. Also, the degree is 4, which is an even number. Therefore, the end behavior is up and up. Now, let's draw the graph!
