If a is a root of `x^(2) - 3x-5=0` find the value of `a^(4) - 2a^(3) - 7a^(2) -8a`

Answer

To solve the problem, we need to find the value of the expression \( a^4 - 2a^3 - 7a^2 - 8a \) given that \( a \) is a root of the equation \( x^2 - 3x - 5 = 0 \).

### Step-by-step Solution:

1. **Identify the quadratic equation**:
The given equation is \( x^2 - 3x - 5 = 0 \). Since \( a \) is a root, we have:
\[ a^2 - 3a - 5 = 0 \]

2. **Express \( a^2 \)**:
From the equation \( a^2 - 3a - 5 = 0 \), we can express \( a^2 \) in terms of \( a \):
\[ a^2 = 3a + 5 \]

3. **Calculate \( a^3 \)**:
To find \( a^3 \), we can multiply both sides of the equation for \( a^2 \) by \( a \):
\[ a^3 = a \cdot a^2 = a(3a + 5) = 3a^2 + 5a \]
Now substitute \( a^2 \) from step 2:
\[ a^3 = 3(3a + 5) + 5a = 9a + 15 + 5a = 14a + 15 \]

4. **Calculate \( a^4 \)**:
To find \( a^4 \), multiply both sides of the equation for \( a^3 \) by \( a \):
\[ a^4 = a \cdot a^3 = a(14a + 15) = 14a^2 + 15a \]
Substitute \( a^2 \) from step 2:
\[ a^4 = 14(3a + 5) + 15a = 42a + 70 + 15a = 57a + 70 \]

5. **Substitute \( a^4 \), \( a^3 \), and \( a^2 \) into the expression**:
Now substitute \( a^4 \), \( a^3 \), and \( a^2 \) into the expression \( a^4 - 2a^3 - 7a^2 - 8a \):
\[ a^4 - 2a^3 - 7a^2 - 8a = (57a + 70) - 2(14a + 15) - 7(3a + 5) - 8a \]

6. **Simplify the expression**:
Calculate each term:
- \( -2(14a + 15) = -28a - 30 \)
- \( -7(3a + 5) = -21a - 35 \)

Now combine all terms:
\[ 57a + 70 - 28a - 30 - 21a - 35 - 8a \]
Combine like terms:
\[ (57a - 28a - 21a - 8a) + (70 - 30 - 35) = 0a + 5 \]

7. **Final result**:
Therefore, the value of \( a^4 - 2a^3 - 7a^2 - 8a \) is:
\[ \boxed{5} \]