Consider the polynomial f`(x)= 1+2x+3x^2+4x^3`. Let s be the sum of all distinct real roots of `f(x)`and let `t= |s|`.

Options

A. `(-(1)/(4),0)`
B. `(-11, -(3)/(4))`
C. `(-(3)/(4), - (1)/(2))`
D. `(0,(1)/(4))`

Answer

To solve the problem, we start with the polynomial given by its derivative:

**Step 1:** Identify the polynomial from the derivative.
Given \( f'(x) = 1 + 2x + 3x^2 + 4x^3 \), we can find \( f(x) \) by integrating \( f'(x) \).

**Step 2:** Integrate \( f'(x) \).
\[ f(x) = \int (1 + 2x + 3x^2 + 4x^3) \, dx = x + x^2 + x^3 + x^4 + C \]
where \( C \) is the constant of integration.

**Step 3:** Find the roots of \( f(x) \).
To find the roots, we set \( f(x) = 0 \):
\[ x + x^2 + x^3 + x^4 + C = 0 \]
Assuming \( C = 0 \) for simplicity (as the constant does not affect the roots), we have:
\[ x + x^2 + x^3 + x^4 = 0 \]
Factoring out \( x \):
\[ x(1 + x + x^2 + x^3) = 0 \]
This gives us one root \( x = 0 \).

**Step 4:** Solve the cubic polynomial.
Now we need to solve \( 1 + x + x^2 + x^3 = 0 \). We can use the Rational Root Theorem or synthetic division to find possible rational roots. Testing \( x = -1 \):
\[ 1 - 1 + 1 - 1 = 0 \]
So \( x = -1 \) is a root. We can factor \( 1 + x + x^2 + x^3 \) as:
\[ (x + 1)(x^2 + 1) = 0 \]
The quadratic \( x^2 + 1 = 0 \) gives us complex roots \( x = i \) and \( x = -i \).

**Step 5:** Identify distinct real roots.
The distinct real roots of \( f(x) \) are \( x = 0 \) and \( x = -1 \).

**Step 6:** Calculate the sum of distinct real roots \( s \).
\[ s = 0 + (-1) = -1 \]

**Step 7:** Calculate \( t = |s| \).
\[ t = |-1| = 1 \]

Thus, the final answers are:
- The sum of all distinct real roots \( s = -1 \)
- The value \( t = |s| = 1 \)