To find all the real zeros of the given polynomial and write the polynomial in factored form, let's break it down step-by-step.
We are given the polynomial:
[tex]\[ P(x) = x^4 - 2x^3 - 48x^2 + 98x - 49 \][/tex]
### Step 1: Find the Zeros
The zeros are the values of [tex]\( x \)[/tex] that make the polynomial equal to zero. According to our factorization:
[tex]\[ P(x) = (x - 7)(x - 1)^2(x + 7) \][/tex]
By setting each factor equal to zero and solving for [tex]\( x \)[/tex], we get the zeros of the polynomial. Let's solve it:
1. [tex]\((x - 7) = 0 \quad \Rightarrow \quad x = 7\)[/tex]
2. [tex]\((x - 1)^2 = 0 \quad \Rightarrow \quad x = 1\)[/tex] (Note: It's a double root because of the squared term)
3. [tex]\((x + 7) = 0 \quad \Rightarrow \quad x = -7\)[/tex]
So, the zeros of the polynomial [tex]\( P(x) \)[/tex] are [tex]\( x = 7 \)[/tex], [tex]\( x = 1 \)[/tex] (with multiplicity 2), and [tex]\( x = -7 \)[/tex].
### Step 2: Write the Polynomial in Factored Form
The factored form combines all the found zeros into the expression based on their multiplicities:
[tex]\[ P(x) = (x - 7)(x - 1)^2(x + 7) \][/tex]
### Summary
The real zeros of the polynomial [tex]\( P(x) = x^4 - 2x^3 - 48x^2 + 98x - 49 \)[/tex] are [tex]\( x = 7 \)[/tex], [tex]\( x = 1 \)[/tex] (with multiplicity 2), and [tex]\( x = -7 \)[/tex].
In factored form, the polynomial is:
[tex]\[ P(x) = (x - 7)(x - 1)^2(x + 7) \][/tex]
Thus, the final answer can be rewritten as follows:
[tex]\[
\boxed{
\begin{array}{l}
\text{Zeros: } x = 7, \, x = 1 \, (\text{multiplicity 2}), \, x = -7 \\
\text{Factored Polynomial: } P(x) = (x - 7)(x - 1)^2(x + 7)
\end{array}
}
\][/tex]
