What must be subtracted from `16x^3- 8x^2+4x+7` so that the resulting expression has 2x+1 is a factor?

Answer

To solve the problem of what must be subtracted from \( 16x^3 - 8x^2 + 4x + 7 \) so that \( 2x + 1 \) is a factor, we can follow these steps:

### Step 1: Identify the polynomial and the factor
Let the polynomial be:
\[ f(x) = 16x^3 - 8x^2 + 4x + 7 \]
We want \( 2x + 1 \) to be a factor of \( f(x) - k \), where \( k \) is the value we need to find.

### Step 2: Find the root of the factor
To find the value of \( k \), we first need to find the root of the factor \( 2x + 1 \):
\[ 2x + 1 = 0 \]
Solving for \( x \):
\[ 2x = -1 \]
\[ x = -\frac{1}{2} \]

### Step 3: Substitute the root into the polynomial
Now we substitute \( x = -\frac{1}{2} \) into the polynomial \( f(x) \):
\[ f\left(-\frac{1}{2}\right) = 16\left(-\frac{1}{2}\right)^3 - 8\left(-\frac{1}{2}\right)^2 + 4\left(-\frac{1}{2}\right) + 7 - k \]

### Step 4: Calculate each term
Calculating each term:
1. \( 16\left(-\frac{1}{2}\right)^3 = 16 \cdot -\frac{1}{8} = -2 \)
2. \( -8\left(-\frac{1}{2}\right)^2 = -8 \cdot \frac{1}{4} = -2 \)
3. \( 4\left(-\frac{1}{2}\right) = -2 \)
4. The constant term is \( 7 \).

Putting it all together:
\[ f\left(-\frac{1}{2}\right) = -2 - 2 - 2 + 7 - k \]

### Step 5: Set the equation to zero
Since \( 2x + 1 \) is a factor, we set the equation to zero:
\[ -2 - 2 - 2 + 7 - k = 0 \]
This simplifies to:
\[ 1 - k = 0 \]

### Step 6: Solve for \( k \)
Solving for \( k \):
\[ k = 1 \]

### Final Answer
Thus, the value that must be subtracted from \( 16x^3 - 8x^2 + 4x + 7 \) so that \( 2x + 1 \) is a factor is:
\[ \boxed{1} \]