To find the roots of the polynomial [tex]\( f(x) = 3x^3 + 12x^2 + 3x - 18 \)[/tex], we must solve the equation [tex]\(3x^3 + 12x^2 + 3x - 18 = 0\)[/tex]. Let's go through the steps to identify the roots:
### Step 1: Identify the Polynomial
The polynomial we are working with is:
[tex]\[ f(x) = 3x^3 + 12x^2 + 3x - 18 \][/tex]
### Step 2: Solve for Roots
To find the roots of the polynomial, we need to find the values of [tex]\(x\)[/tex] that satisfy the equation [tex]\(3x^3 + 12x^2 + 3x - 18 = 0\)[/tex]. These are the points where the polynomial intersects the x-axis.
After solving the polynomial equation, we find the roots to be:
[tex]\[ x = -3, x = -2, \text{ and } x = 1 \][/tex]
### Step 3: Determine the Largest Root
Among the roots obtained:
[tex]\[ -3, -2, 1 \][/tex]
The largest value is:
[tex]\[ 1 \][/tex]
### Conclusion
The roots of the polynomial [tex]\( f(x) = 3x^3 + 12x^2 + 3x - 18 \)[/tex] are [tex]\( x = -3 \)[/tex], [tex]\( x = -2 \)[/tex], and [tex]\( x = 1 \)[/tex]. The largest root is:
[tex]\[ 1 \][/tex]
