Answer:
To find the zeros of the function \( f(x) = 4x^4 - 16x^3 - 8x^2 + 16x + 64 \), we need to find the values of \( x \) where \( f(x) \) equals zero.
a. **Finding Zeros**:
To find the zeros of the function, we set \( f(x) \) equal to zero and solve for \( x \):
\[ 4x^4 - 16x^3 - 8x^2 + 16x + 64 = 0 \]
We can factor out a common factor of \( 4 \) from all terms:
\[ 4(x^4 - 4x^3 - 2x^2 + 4x + 16) = 0 \]
Then, we can apply various methods to find the roots, such as factoring, synthetic division, or numerical methods like the Newton-Raphson method.
b. **Drawing the Graph**:
To sketch the graph without a graphing utility, we can analyze the behavior of the polynomial function by looking at its leading term and the behavior around the zeros.
- **Leading Term**: The leading term of \( f(x) \) is \( 4x^4 \), indicating that the function approaches positive infinity as \( x \) approaches positive or negative infinity.
- **Zeros**: We found the zeros in part a.
- **Behavior around Zeros**: We can examine the sign of \( f(x) \) in intervals between the zeros to sketch the graph accurately.
With this information, we can sketch a complete graph of the function by plotting the zeros and understanding how the function behaves between these zeros. We would also consider the end behavior, which is that the graph approaches positive infinity as \( x \) approaches positive or negative infinity.
