Questions: Division (Due Question 7, 3.2.45 Use the remainder theorem and synthetic division to find f(k) for the given value of k. f(x)=4x^4-23x^3-2x^2+5x+5; k=-1/4 f(-1/4)=

Solution Steps

To find \( f(k) \) using the remainder theorem and synthetic division, we first set up synthetic division with the polynomial coefficients and the value of \( k \). The remainder of this division will give us \( f(k) \).

We are given the polynomial \( f(x) = 4x^4 - 23x^3 - 2x^2 + 5x + 5 \) and need to evaluate it at \( k = -\frac{1}{4} \).

Using synthetic division with the coefficients \( [4, -23, -2, 5, 5] \) and \( k = -\frac{1}{4} \), we perform the following calculations:

1. Start with the leading coefficient: \( 4 \).
2. Multiply by \( k \) and add the next coefficient:
   • \( 4 \times -\frac{1}{4} + (-23) = 0 - 23 = -23 \)
3. Repeat for the next coefficients:
   • \( -23 \times -\frac{1}{4} + (-2) = \frac{23}{4} - 2 = \frac{23}{4} - \frac{8}{4} = \frac{15}{4} \)
   • \( \frac{15}{4} \times -\frac{1}{4} + 5 = -\frac{15}{16} + 5 = -\frac{15}{16} + \frac{80}{16} = \frac{65}{16} \)
   • \( \frac{65}{16} \times -\frac{1}{4} + 5 = -\frac{65}{64} + 5 = -\frac{65}{64} + \frac{320}{64} = \frac{255}{64} \)

The final result from synthetic division gives us \( f\left(-\frac{1}{4}\right) = 4.0 \).

Final Answer