Let's work through the problem step-by-step to find the real zeros of the polynomial [tex]\(f(x)\)[/tex] and use those zeros to factor [tex]\(f\)[/tex].
Given:
[tex]\[ f(x) = 3x^3 - 43x^2 + 161x - 49 \][/tex]
### Step 1: Find the real zeros of [tex]\(f(x)\)[/tex]
To find the real zeros of the polynomial [tex]\(f(x)\)[/tex], we need to find the values of [tex]\(x\)[/tex] where [tex]\(f(x) = 0\)[/tex].
### Step 2: Solving [tex]\(f(x) = 0\)[/tex]
The real solutions for [tex]\(f(x) = 0\)[/tex] are:
[tex]\[ x = \frac{1}{3} \quad \text{and} \quad x = 7 \][/tex]
Thus, the correct choice for the real zeros is:
A. The real zero(s) of [tex]\(f\)[/tex] is/are [tex]\(x = \frac{1}{3}, 7\)[/tex].
### Step 3: Use the real zeros to factor [tex]\(f\)[/tex]
Given the real zeros [tex]\(x = \frac{1}{3}\)[/tex] and [tex]\(x = 7\)[/tex], we can use these zeros to factor our polynomial.
#### Factoring Process:
- If [tex]\(x = \frac{1}{3}\)[/tex] is a root, then [tex]\((3x - 1)\)[/tex] is a factor.
- If [tex]\(x = 7\)[/tex] is a root, then [tex]\((x - 7)\)[/tex] is a factor.
Since [tex]\(x = 7\)[/tex] is a repeated root, [tex]\((x - 7)\)[/tex] appears twice in the factors of [tex]\(f(x)\)[/tex].
So, the factored form of [tex]\(f(x)\)[/tex] becomes:
[tex]\[ f(x) = (x - 7)^2 (3x - 1) \][/tex]
### Final Answer:
The completely factored form of the polynomial [tex]\(f(x)\)[/tex] is:
[tex]\[
f(x) = (x - 7)^2 (3x - 1)
\][/tex]
