`g(x) =x^4-kx^3+13x^2-12x+4`
The function g is defined above, and k is a constant. In the xy- plane, the graph of g intersects the y - axis at `(0, 4)` and intersects the x - axis at `(1, 0)` and `(2, 0)`. What is the value of k?

Answer

To find the value of \( k \) in the function \( g(x) = x^4 - kx^3 + 13x^2 - 12x + 4 \), we will use the information given about the intersections of the graph with the axes.

### Step 1: Use the x-intercept at (1, 0)
Since the graph intersects the x-axis at \( (1, 0) \), this means that \( g(1) = 0 \). We can substitute \( x = 1 \) into the function:

\[ g(1) = 1^4 - k(1^3) + 13(1^2) - 12(1) + 4 \]

### Step 2: Simplify the equation
Calculating each term, we have:

\[ g(1) = 1 - k + 13 - 12 + 4 \]

Combining these values gives:

\[ g(1) = 1 - k + 5 = 6 - k \]

### Step 3: Set the equation to zero
Since \( g(1) = 0 \), we set the equation to zero:

\[ 6 - k = 0 \]

### Step 4: Solve for \( k \)
Rearranging the equation gives:

\[ k = 6 \]

### Final Answer
Thus, the value of \( k \) is:

\[ \boxed{6} \]

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