Questions: Use synthetic division to decide whether the given number k is a zero of the polynomial function. If it is not, give the value of f(k) f(x)=2x^3-11x^2-3x+56; k=4 Is 4 a zero of the function? Select the correct choice below and, if necessary, fill in the answer box to complete your choice A. The given k is not a zero of the polynomial function f(4)= B. The given k is a zero of the polynomial function.

Solution Steps

To determine if \( k = 4 \) is a zero of the polynomial function \( f(x) = 2x^3 - 11x^2 - 3x + 56 \), we can use synthetic division. If the remainder is zero, then \( k \) is a zero of the polynomial. Otherwise, we will find \( f(4) \).

1. Set up the synthetic division with \( k = 4 \) and the coefficients of the polynomial \( f(x) \).
2. Perform the synthetic division process.
3. Check the remainder:
   • If the remainder is zero, \( k \) is a zero of the polynomial.
   • If the remainder is not zero, calculate \( f(4) \).

We are given the polynomial function \( f(x) = 2x^3 - 11x^2 - 3x + 56 \) and need to determine if \( k = 4 \) is a zero of this polynomial. We will perform synthetic division using the coefficients \( [2, -11, -3, 56] \).

Using synthetic division with \( k = 4 \):

• Start with the leading coefficient: \( 2 \).
• Multiply \( 2 \) by \( 4 \) to get \( 8 \) and add it to \( -11 \) to get \( -3 \).
• Multiply \( -3 \) by \( 4 \) to get \( -12 \) and add it to \( -3 \) to get \( -15 \).
• Multiply \( -15 \) by \( 4 \) to get \( -60 \) and add it to \( 56 \) to get \( -4 \).

The result of the synthetic division is \( [2, -3, -15, -4] \), where \( -4 \) is the remainder.

Since the remainder is \( -4 \), we conclude that \( k = 4 \) is not a zero of the polynomial function. We can also express this as \( f(4) = -4 \).

Final Answer