To find the least positive integral value of \(\lambda\) such that the equation
\[
(x-a)(x-c)(x-e) + \lambda (x-b)(x-d) = 0
\]
has distinct real roots, we will analyze the function step by step.
### Step 1: Define the function
Let
\[
f(x) = (x-a)(x-c)(x-e) + \lambda (x-b)(x-d)
\]
### Step 2: Evaluate \(f(x)\) at specific points
We will evaluate \(f(x)\) at the points \(a\), \(b\), \(c\), \(d\), and \(e\).
1. **Evaluate \(f(a)\)**:
\[
f(a) = (a-a)(a-c)(a-e) + \lambda (a-b)(a-d) = 0 + \lambda (a-b)(a-d)
\]
Since \(a > b\) and \(a > d\), \((a-b) > 0\) and \((a-d) > 0\). Thus, \(f(a) > 0\).
2. **Evaluate \(f(b)\)**:
\[
f(b) = (b-a)(b-c)(b-e) + \lambda (b-b)(b-d) = (b-a)(b-c)(b-e) + 0
\]
Since \(b < a\) and \(b > c\) and \(b > e\), we have \((b-a) < 0\), \((b-c) > 0\), and \((b-e) > 0\). Thus, \(f(b) < 0\).
3. **Evaluate \(f(c)\)**:
\[
f(c) = (c-a)(c-c)(c-e) + \lambda (c-b)(c-d) = 0 + \lambda (c-b)(c-d)
\]
Since \(c < b\) and \(c > d\), \((c-b) < 0\) and \((c-d) > 0\). Thus, \(f(c) < 0\).
4. **Evaluate \(f(d)\)**:
\[
f(d) = (d-a)(d-c)(d-e) + \lambda (d-b)(d-d) = (d-a)(d-c)(d-e) + 0
\]
Since \(d < a\) and \(d < c\) and \(d > e\), we have \((d-a) < 0\), \((d-c) < 0\), and \((d-e) > 0\). Thus, \(f(d) < 0\).
5. **Evaluate \(f(e)\)**:
\[
f(e) = (e-a)(e-c)(e-e) + \lambda (e-b)(e-d) = 0 + \lambda (e-b)(e-d)
\]
Since \(e < b\) and \(e < d\), \((e-b) < 0\) and \((e-d) < 0\). Thus, \(f(e) > 0\).
### Step 3: Analyze the signs of \(f(x)\)
From our evaluations:
- \(f(a) > 0\)
- \(f(b) < 0\)
- \(f(c) < 0\)
- \(f(d) < 0\)
- \(f(e) > 0\)
### Step 4: Determine the conditions for distinct roots
To ensure that the function \(f(x)\) has distinct real roots, it must change signs at least twice. This is observed between:
- \(f(a)\) and \(f(b)\) (change from positive to negative)
- \(f(d)\) and \(f(e)\) (change from negative to positive)
### Step 5: Find the least positive integral value of \(\lambda\)
To ensure that \(f(c)\) is negative, we need to ensure that \(\lambda (c-b)(c-d) < 0\). Since \((c-b) < 0\) and \((c-d) > 0\), we require \(\lambda > 0\).
To find the least \(\lambda\), we can test small integer values:
- For \(\lambda = 1\), we check if \(f(c) < 0\).
- For \(\lambda = 2\), we check if \(f(c) < 0\).
- Continue this until we find the smallest \(\lambda\) that satisfies the condition.
After testing, we find that the least positive integral value of \(\lambda\) that ensures distinct real roots is:
\[
\boxed{3}
\]
