The polynomial function p is defined by `p(x)=4x^(3)+bx^(2)+41x+12`, where b is a constant. When graphed on a standard coordinate plane, p intersects the x-axis at `(-0.25, 0), (3, 0) and (k, 0)`. What is the value of b?

Options

A. `-27`
B. `-sqrt(207)`
C. `4`
D. `27`

Explanation

To find the value of \( b \) in the polynomial function \( p(x) = 4x^3 + bx^2 + 41x + 12 \), given that it intersects the x-axis at the points \( (-0.25, 0) \), \( (3, 0) \), and \( (k, 0) \), we can follow these steps:

### Step 1: Substitute the x-value of one intersection point into the polynomial
We will use the point \( (3, 0) \) for our calculations since it is an integer and easier to work with. We substitute \( x = 3 \) into the polynomial:

\[ p(3) = 4(3)^3 + b(3)^2 + 41(3) + 12 \]

### Step 2: Calculate \( p(3) \)
Calculating each term:

- \( 4(3)^3 = 4 \times 27 = 108 \)
- \( b(3)^2 = 9b \)
- \( 41(3) = 123 \)
- The constant term is \( 12 \)

Putting it all together:

\[ p(3) = 108 + 9b + 123 + 12 \]

### Step 3: Set the equation to zero
Since \( p(3) = 0 \) (because it intersects the x-axis at this point), we set the equation to zero:

\[ 108 + 9b + 123 + 12 = 0 \]

### Step 4: Simplify the equation
Combine the constant terms:

\[ 108 + 123 + 12 = 243 \]

So, we have:

\[ 243 + 9b = 0 \]

### Step 5: Solve for \( b \)
Now, isolate \( b \):

\[ 9b = -243 \]
\[ b = \frac{-243}{9} = -27 \]

### Conclusion
The value of \( b \) is \( -27 \).

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