Absolutely, let's factorize the given polynomial step-by-step:
Given polynomial:
[tex]\[ x^3 - 3x^2 - 4x + 12 \][/tex]
### Step 1: Identify potential rational roots
The Rational Root Theorem states that any rational root of the polynomial [tex]\( p(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0 \)[/tex] is a fraction [tex]\( \frac{p}{q} \)[/tex] where [tex]\( p \)[/tex] is a factor of the constant term [tex]\( a_0 \)[/tex] and [tex]\( q \)[/tex] is a factor of the leading coefficient [tex]\( a_n \)[/tex].
For the polynomial [tex]\( x^3 - 3x^2 - 4x + 12 \)[/tex]:
- The constant term [tex]\( a_0 = 12 \)[/tex]
- The leading coefficient [tex]\( a_n = 1 \)[/tex]
Potential rational roots are the factors of 12: [tex]\( \pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 12 \)[/tex].
### Step 2: Test the potential rational roots
Substitute each potential root into the polynomial to see if it equals zero.
1. Test [tex]\( x = 1 \)[/tex]:
[tex]\[
1^3 - 3(1)^2 - 4(1) + 12 = 1 - 3 - 4 + 12 = 6 \neq 0
\][/tex]
So, [tex]\( x = 1 \)[/tex] is not a root.
2. Test [tex]\( x = 2 \)[/tex]:
[tex]\[
2^3 - 3(2)^2 - 4(2) + 12 = 8 - 12 - 8 + 12 = 0
\][/tex]
So, [tex]\( x = 2 \)[/tex] is a root.
### Step 3: Perform polynomial division
Since [tex]\( x = 2 \)[/tex] is a root, we can factor [tex]\( (x - 2) \)[/tex] out of the polynomial [tex]\( x^3 - 3x^2 - 4x + 12 \)[/tex].
Use synthetic division to divide the polynomial by [tex]\( (x - 2) \)[/tex]:
[tex]\[
\begin{array}{r|rrrr}
2 & 1 & -3 & -4 & 12 \\
& & 2 & -2 & -12 \\
\hline
& 1 & -1 & -6 & 0 \\
\end{array}
\][/tex]
The result of synthetic division is:
[tex]\[ x^2 - x - 6 \][/tex]
### Step 4: Factor the quadratic polynomial
Next, we need to factorize [tex]\( x^2 - x - 6 \)[/tex]. We look for two numbers that multiply to [tex]\(-6\)[/tex] and add to [tex]\(-1\)[/tex]. These numbers are [tex]\(-3\)[/tex] and [tex]\(2\)[/tex].
So, we can write:
[tex]\[ x^2 - x - 6 = (x - 3)(x + 2) \][/tex]
### Step 5: Combine the factors
Now, combining all the factors we obtain:
[tex]\[ x^3 - 3x^2 - 4x + 12 = (x - 2)(x - 3)(x + 2) \][/tex]
### Final Answer:
The factorization of the polynomial [tex]\( x^3 - 3x^2 - 4x + 12 \)[/tex] is:
[tex]\[
(x - 2)(x - 3)(x + 2)
\][/tex]
