`(y^(2)+ky-3)(y-4)=y^(3)+by^(2)+5y+12`
In the equation above, k is a nonzero constant. If the equation is true for all values of y, what is the value of k?

Options

A. `-(1)/(2)`
B. `-2`
C. `4`
D. `6`

Answer

To solve the equation \((y^{2}+ky-3)(y-4)=y^{3}+by^{2}+5y+12\) for the value of \(k\), we will follow these steps:

### Step 1: Expand the Left-Hand Side
We start by expanding the left-hand side of the equation:

\[ (y^{2} + ky - 3)(y - 4) \]

Distributing each term in the first polynomial by each term in the second polynomial:

1. Multiply \(y^{2}\) by \(y\):
\[ y^{2} \cdot y = y^{3} \]

2. Multiply \(y^{2}\) by \(-4\):
\[ y^{2} \cdot (-4) = -4y^{2} \]

3. Multiply \(ky\) by \(y\):
\[ ky \cdot y = ky^{2} \]

4. Multiply \(ky\) by \(-4\):
\[ ky \cdot (-4) = -4ky \]

5. Multiply \(-3\) by \(y\):
\[ -3 \cdot y = -3y \]

6. Multiply \(-3\) by \(-4\):
\[ -3 \cdot (-4) = 12 \]

Now, combine all these results:
\[ y^{3} + (ky^{2} - 4y^{2}) + (-4ky - 3y) + 12 \]

### Step 2: Combine Like Terms
Combining the like terms gives us:
\[ y^{3} + (k - 4)y^{2} + (-4k - 3)y + 12 \]

### Step 3: Set the Expanded Form Equal to the Right-Hand Side
Now we set the expanded left-hand side equal to the right-hand side of the equation:
\[ y^{3} + (k - 4)y^{2} + (-4k - 3)y + 12 = y^{3} + by^{2} + 5y + 12 \]

### Step 4: Compare Coefficients
Since the equation is true for all values of \(y\), we can compare the coefficients of corresponding powers of \(y\).

1. For \(y^{2}\):
\[ k - 4 = b \]

2. For \(y\):
\[ -4k - 3 = 5 \]

### Step 5: Solve for \(k\)
From the equation \(-4k - 3 = 5\):
\[ -4k = 5 + 3 \]
\[ -4k = 8 \]
\[ k = -2 \]

### Step 6: Conclusion
Thus, the value of \(k\) is:
\[ \boxed{-2} \]