The repeated root of the equation `4x^3 -12x^2 -15x -4=0` is

Options

A. `5/2`
B. `-1/2`
C. `1/3`
D. `-1/3`

Answer

To find the repeated root of the cubic equation \(4x^3 - 12x^2 - 15x - 4 = 0\), we will follow these steps:

### Step 1: Identify Possible Rational Roots
Using the Rational Root Theorem, we can identify possible rational roots. The possible rational roots are the factors of the constant term (-4) divided by the factors of the leading coefficient (4). The factors of -4 are \(\pm 1, \pm 2, \pm 4\) and the factors of 4 are \(\pm 1, \pm 2, \pm 4\). Thus, the possible rational roots are:
\[ \pm 1, \pm \frac{1}{2}, \pm 2, \pm \frac{1}{4}, \pm 4 \]

### Step 2: Test Possible Roots
We will test these possible roots in the polynomial \(f(x) = 4x^3 - 12x^2 - 15x - 4\) to see which one is a root.

Testing \(x = 4\):
\[ f(4) = 4(4)^3 - 12(4)^2 - 15(4) - 4 = 4(64) - 12(16) - 60 - 4 = 256 - 192 - 60 - 4 = 0 \]
Thus, \(x = 4\) is a root.

### Step 3: Factor the Polynomial
Since \(x = 4\) is a root, we can factor \(f(x)\) as:
\[ f(x) = (x - 4)(\text{quadratic factor}) \]
To find the quadratic factor, we perform polynomial long division of \(f(x)\) by \(x - 4\).

### Step 4: Perform Polynomial Long Division
Dividing \(4x^3 - 12x^2 - 15x - 4\) by \(x - 4\):
1. Divide the leading term: \(4x^3 \div x = 4x^2\).
2. Multiply \(4x^2\) by \(x - 4\): \(4x^3 - 16x^2\).
3. Subtract:
\[ (-12x^2 + 16x^2) - 15x - 4 = 4x^2 - 15x - 4 \]
4. Divide the leading term: \(4x^2 \div x = 4x\).
5. Multiply \(4x\) by \(x - 4\): \(4x^2 - 16x\).
6. Subtract:
\[ (-15x + 16x) - 4 = x - 4 \]
7. Divide the leading term: \(x \div x = 1\).
8. Multiply \(1\) by \(x - 4\): \(x - 4\).
9. Subtract:
\[ 0 \]

Thus, we have:
\[ f(x) = (x - 4)(4x^2 + 4x + 1) \]

### Step 5: Solve the Quadratic Equation
To find the repeated roots, we need to solve \(4x^2 + 4x + 1 = 0\). We can use the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
where \(a = 4\), \(b = 4\), and \(c = 1\):
\[ x = \frac{-4 \pm \sqrt{4^2 - 4 \cdot 4 \cdot 1}}{2 \cdot 4} = \frac{-4 \pm \sqrt{16 - 16}}{8} = \frac{-4 \pm 0}{8} = \frac{-4}{8} = -\frac{1}{2} \]

### Conclusion
The repeated root of the equation \(4x^3 - 12x^2 - 15x - 4 = 0\) is:
\[ \boxed{-\frac{1}{2}} \]