Answer :

To find the roots of the polynomial [tex]\( f(x) = x^3 + 2x^2 - 11x - 12 \)[/tex], we need to identify the values of [tex]\( x \)[/tex] that make [tex]\( f(x) = 0 \)[/tex]. Here are the steps for solving it:

1. Identify the polynomial: We start with the polynomial equation [tex]\( f(x) = x^3 + 2x^2 - 11x - 12 \)[/tex].

2. Factor the polynomial: We look for factors of the polynomial that, when multiplied, give the original polynomial. To do this, we can use methods like synthetic division, the Rational Root Theorem, or factoring by grouping if feasible.

3. Find the rational roots: Attempting possible rational roots (which are the factors of the constant term [tex]\((-12)\)[/tex] divided by the factors of the leading coefficient ([tex]\(1\)[/tex])):
- Factors of [tex]\(-12\)[/tex]: [tex]\( \pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 12 \)[/tex]
- By testing these possible roots, we find that:
- [tex]\( x = 3 \)[/tex] is a root because substituting [tex]\( x = 3 \)[/tex] into the polynomial gives [tex]\( 0 \)[/tex].

4. Divide the polynomial by [tex]\( (x - 3) \)[/tex]: Since [tex]\( x = 3 \)[/tex] is a root, we can factor out [tex]\( (x - 3) \)[/tex] from the polynomial. Performing synthetic division or polynomial division gives:
[tex]\[ x^3 + 2x^2 - 11x - 12 = (x - 3)(x^2 + 5x + 4) \][/tex]

5. Factor the quadratic polynomial [tex]\( x^2 + 5x + 4 \)[/tex]:
[tex]\[ x^2 + 5x + 4 = (x + 4)(x + 1) \][/tex]

6. Combine all factors to express the polynomial: Now we have the polynomial fully factored:
[tex]\[ x^3 + 2x^2 - 11x - 12 = (x - 3)(x + 4)(x + 1) \][/tex]

7. Find all the roots: Setting each factor equal to zero gives:
[tex]\[ x - 3 = 0 \implies x = 3 \][/tex]
[tex]\[ x + 4 = 0 \implies x = -4 \][/tex]
[tex]\[ x + 1 = 0 \implies x = -1 \][/tex]

So, the roots of the polynomial are [tex]\( x = 3 \)[/tex], [tex]\( x = -4 \)[/tex], and [tex]\( x = -1 \)[/tex].

8. Identify the largest root: Among the roots [tex]\( -4, -1, 3 \)[/tex], the largest root is [tex]\( 3 \)[/tex].

Therefore, the roots of the polynomial are [tex]\(-4\)[/tex], [tex]\(-1\)[/tex], and [tex]\(3\)[/tex]. The value of the largest root is [tex]\(3\)[/tex].