Questions: Find the zeros and their multiplicities. Consider using Descartes' rule of signs and the upper and lower bound theorem to limit your search for rational zeros. f(x)=4x^3-16x^2+9x+9 Part 1 of 2 If there is more than one answer, separate them with commas. Select "None" if applicable. The zeros of f(x):-1/2, 3/2, 3. Part: 1 / 2 Part 2 of 2 The multiplicity of 3 is . The multiplicity of -1/2 is The multiplicity of 3/2 is

Solution Steps

To find the zeros and their multiplicities of the polynomial \( f(x) = 4x^3 - 16x^2 + 9x + 9 \), we can use the following approach:

1. Descartes' Rule of Signs: This will help us determine the possible number of positive and negative real zeros.
2. Rational Root Theorem: This will help us identify possible rational zeros by considering factors of the constant term and the leading coefficient.
3. Synthetic Division: Use this to test potential zeros and factor the polynomial.
4. Multiplicity: Once the zeros are found, determine their multiplicities by examining the factored form of the polynomial.

The polynomial \( f(x) = 4x^3 - 16x^2 + 9x + 9 \) has been analyzed, and the zeros have been determined to be:

• \( x = 3 \)
• \( x = \frac{3}{2} \)
• \( x = -\frac{1}{2} \)

From the analysis, each zero has a multiplicity of 1. Therefore, we have:

• The multiplicity of \( 3 \) is \( 1 \).
• The multiplicity of \( \frac{3}{2} \) is \( 1 \).
• The multiplicity of \( -\frac{1}{2} \) is \( 1 \).

Final Answer