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To determine the possible rational zeros of the polynomial \( f(x) = 10x^3 + 9x^2 - 9x + 4 \), we can use the Rational Zeros Theorem. Here's a step-by-step solution to find possible rational zeros:
### Step 1: Identifying Coefficients
First, identify the coefficients of the polynomial. For \( f(x) = 10x^3 + 9x^2 - 9x + 4 \):
- The constant term (\( a_0 \)) is 4.
- The leading coefficient (\( a_n \)) is 10.
### Step 2: Factors of the Constant Term (\( a_0 \))
List all factors of the constant term 4:
- Positive factors: 1, 2, 4
- Negative factors: -1, -2, -4
Thus, the set of factors for \( a_0 \) is: \( \{1, 2, 4, -1, -2, -4\} \).
### Step 3: Factors of the Leading Coefficient (\( a_n \))
List all factors of the leading coefficient 10:
- Positive factors: 1, 2, 5, 10
- Negative factors: -1, -2, -5, -10
Thus, the set of factors for \( a_n \) is: \( \{1, 2, 5, 10, -1, -2, -5, -10\} \).
### Step 4: Forming Possible Rational Zeros
The Rational Zeros Theorem states that the possible rational zeros of the polynomial are given by \( \frac{p}{q} \), where \( p \) is a factor of the constant term, and \( q \) is a factor of the leading coefficient.
Thus, we form all possible combinations of \( \frac{p}{q} \):
- Positive combinations: \( 1, 2, 4, \frac{2}{2} = 1, \frac{4}{2} = 2, \frac{1}{5} = 0.2, \frac{2}{5} = 0.4, \frac{4}{5} = 0.8, \frac{1}{10} = 0.1, \frac{2}{10} = 0.2, \frac{4}{10} = 0.4 \)
- Negative combinations: \(-1, -2, -4, \frac{-2}{2} = -1, \frac{-4}{2} = -2, \frac{-1}{5} = -0.2, \frac{-2}{5} = -0.4, \frac{-4}{5} = -0.8, \frac{-1}{10} = -0.1, \frac{-2}{10} = -0.2, \frac{-4}{10} = -0.4\)
After removing the duplicates, we get:
- Positive rational zeros: 1, 2, 4, 0.2, 0.4, 0.8, 0.1, 0.5
- Negative rational zeros: -1, -2, -4, -0.2, -0.4, -0.8, -0.1, -0.5
### Step 5: Listing All Possible Rational Zeros
Combining positive and negative rational zeros, we obtain the complete list:
[tex]\[ \{-4.0, -2.0, -1.0, -0.8, -0.5, -0.4, -0.2, -0.1, 0.1, 0.2, 0.4, 0.5, 0.8, 1.0, 2.0, 4.0\} \][/tex]
There should be no duplicates, and the list is typically presented in ascending order:
### Final Answer
The list of all possible rational zeros of the polynomial \( f(x) = 10x^3 + 9x^2 - 9x + 4 \) is:
[tex]\[ \{-4.0, -2.0, -1.0, -0.8, -0.5, -0.4, -0.2, -0.1, 0.1, 0.2, 0.4, 0.5, 0.8, 1.0, 2.0, 4.0\} \][/tex]
### Step 1: Identifying Coefficients
First, identify the coefficients of the polynomial. For \( f(x) = 10x^3 + 9x^2 - 9x + 4 \):
- The constant term (\( a_0 \)) is 4.
- The leading coefficient (\( a_n \)) is 10.
### Step 2: Factors of the Constant Term (\( a_0 \))
List all factors of the constant term 4:
- Positive factors: 1, 2, 4
- Negative factors: -1, -2, -4
Thus, the set of factors for \( a_0 \) is: \( \{1, 2, 4, -1, -2, -4\} \).
### Step 3: Factors of the Leading Coefficient (\( a_n \))
List all factors of the leading coefficient 10:
- Positive factors: 1, 2, 5, 10
- Negative factors: -1, -2, -5, -10
Thus, the set of factors for \( a_n \) is: \( \{1, 2, 5, 10, -1, -2, -5, -10\} \).
### Step 4: Forming Possible Rational Zeros
The Rational Zeros Theorem states that the possible rational zeros of the polynomial are given by \( \frac{p}{q} \), where \( p \) is a factor of the constant term, and \( q \) is a factor of the leading coefficient.
Thus, we form all possible combinations of \( \frac{p}{q} \):
- Positive combinations: \( 1, 2, 4, \frac{2}{2} = 1, \frac{4}{2} = 2, \frac{1}{5} = 0.2, \frac{2}{5} = 0.4, \frac{4}{5} = 0.8, \frac{1}{10} = 0.1, \frac{2}{10} = 0.2, \frac{4}{10} = 0.4 \)
- Negative combinations: \(-1, -2, -4, \frac{-2}{2} = -1, \frac{-4}{2} = -2, \frac{-1}{5} = -0.2, \frac{-2}{5} = -0.4, \frac{-4}{5} = -0.8, \frac{-1}{10} = -0.1, \frac{-2}{10} = -0.2, \frac{-4}{10} = -0.4\)
After removing the duplicates, we get:
- Positive rational zeros: 1, 2, 4, 0.2, 0.4, 0.8, 0.1, 0.5
- Negative rational zeros: -1, -2, -4, -0.2, -0.4, -0.8, -0.1, -0.5
### Step 5: Listing All Possible Rational Zeros
Combining positive and negative rational zeros, we obtain the complete list:
[tex]\[ \{-4.0, -2.0, -1.0, -0.8, -0.5, -0.4, -0.2, -0.1, 0.1, 0.2, 0.4, 0.5, 0.8, 1.0, 2.0, 4.0\} \][/tex]
There should be no duplicates, and the list is typically presented in ascending order:
### Final Answer
The list of all possible rational zeros of the polynomial \( f(x) = 10x^3 + 9x^2 - 9x + 4 \) is:
[tex]\[ \{-4.0, -2.0, -1.0, -0.8, -0.5, -0.4, -0.2, -0.1, 0.1, 0.2, 0.4, 0.5, 0.8, 1.0, 2.0, 4.0\} \][/tex]
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