We want to graph the given polynomial function.
h(x)=(x-3)(x^2+x+1)
Let's start by finding the zeros of the function.
Zeros of the Function
To do so, we need to find the values of x for which h(x)=0.
h(x)=0 ⇔ (x-3)(x^2+x+1)=0
Since the function is already written in factored form, we will use the Zero Product Property. Note that the second part does not factor out anymore and has no real zeros, so we will omit it.
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
(x-3)(x^2+x+1)
h(x)=(x-3)(x^2+x+1)
- 1
( - 1-3)(( - 1)^2+( - 1)+1)
- 4
0
( 0-3)( 0^2+ 0+1)
- 3
1
( 1-3)( 1^2+ 1+1)
- 6
2
( 2-3)( 2^2+ 2+1)
-7
The points ( - 1, -4), ( 0, -3), ( 1, -6), and ( 2, -7) are on the graph of the function. Finally, let's apply the Distributive Property. This will simplify the equation and determine the leading coefficient and degree of the polynomial function.
We can see now that the leading coefficient is 1, which is a positive number. Also, the degree is 3, which is an odd number. Therefore, the end behavior is down and up. Now, let's draw the graph!
