Answer :
To find all possible rational zeros of the polynomial \( h(x) = -7x^3 + 4x^2 - 8x - 3 \) using the Rational Zeros Theorem, we'll follow these steps:
### Step 1: Identify the Constant Term and Leading Coefficient
- The constant term of the polynomial \( h(x) \) is \( -3 \).
- The leading coefficient of the polynomial \( h(x) \) is \( -7 \).
### Step 2: List All Factors of the Constant Term (\(p\))
- Possible factors of \(-3\) are \( \pm 1, \pm 3 \).
### Step 3: List All Factors of the Leading Coefficient (\(q\))
- Possible factors of \(-7\) are \( \pm 1, \pm 7 \).
### Step 4: Form All Possible Ratios \( \frac{p}{q} \)
According to the Rational Zeros Theorem, any rational zero of the polynomial will be a ratio \( \frac{p}{q} \), where \( p \) is a factor of the constant term and \( q \) is a factor of the leading coefficient.
Thus, we consider all pairs:
- \( \frac{1}{1}, \frac{1}{-1}, \frac{1}{7}, \frac{1}{-7} \)
- \( \frac{-1}{1}, \frac{-1}{-1}, \frac{-1}{7}, \frac{-1}{-7} \)
- \( \frac{3}{1}, \frac{3}{-1}, \frac{3}{7}, \frac{3}{-7} \)
- \( \frac{-3}{1}, \frac{-3}{-1}, \frac{-3}{7}, \frac{-3}{-7} \)
### Step 5: Simplify Each Ratio and Remove Duplicates
After simplifying all ratios, we combine and remove duplicates from the list to get the final set of possible rational zeros:
[tex]\[ \left\{ \pm 1, \pm 3, \pm \frac{1}{7}, \pm \frac{3}{7} \right\} \][/tex]
### Step 6: Convert to Decimal Form for Clarity (Optional)
For convenience, we may present the possible rational zeros in decimal form:
[tex]\[ -3.0, -1.0, -0.42857142857142855, -0.14285714285714285, 0.14285714285714285, 0.42857142857142855, 1.0, 3.0 \][/tex]
### Conclusion
Listing all possible rational zeros, we obtain the following values:
[tex]\[ -3.0, -1.0, -0.42857142857142855, -0.14285714285714285, 0.14285714285714285, 0.42857142857142855, 1.0, 3.0 \][/tex]
These are the potential rational solutions for the given polynomial [tex]\( h(x) = -7x^3 + 4x^2 - 8x - 3 \)[/tex].
### Step 1: Identify the Constant Term and Leading Coefficient
- The constant term of the polynomial \( h(x) \) is \( -3 \).
- The leading coefficient of the polynomial \( h(x) \) is \( -7 \).
### Step 2: List All Factors of the Constant Term (\(p\))
- Possible factors of \(-3\) are \( \pm 1, \pm 3 \).
### Step 3: List All Factors of the Leading Coefficient (\(q\))
- Possible factors of \(-7\) are \( \pm 1, \pm 7 \).
### Step 4: Form All Possible Ratios \( \frac{p}{q} \)
According to the Rational Zeros Theorem, any rational zero of the polynomial will be a ratio \( \frac{p}{q} \), where \( p \) is a factor of the constant term and \( q \) is a factor of the leading coefficient.
Thus, we consider all pairs:
- \( \frac{1}{1}, \frac{1}{-1}, \frac{1}{7}, \frac{1}{-7} \)
- \( \frac{-1}{1}, \frac{-1}{-1}, \frac{-1}{7}, \frac{-1}{-7} \)
- \( \frac{3}{1}, \frac{3}{-1}, \frac{3}{7}, \frac{3}{-7} \)
- \( \frac{-3}{1}, \frac{-3}{-1}, \frac{-3}{7}, \frac{-3}{-7} \)
### Step 5: Simplify Each Ratio and Remove Duplicates
After simplifying all ratios, we combine and remove duplicates from the list to get the final set of possible rational zeros:
[tex]\[ \left\{ \pm 1, \pm 3, \pm \frac{1}{7}, \pm \frac{3}{7} \right\} \][/tex]
### Step 6: Convert to Decimal Form for Clarity (Optional)
For convenience, we may present the possible rational zeros in decimal form:
[tex]\[ -3.0, -1.0, -0.42857142857142855, -0.14285714285714285, 0.14285714285714285, 0.42857142857142855, 1.0, 3.0 \][/tex]
### Conclusion
Listing all possible rational zeros, we obtain the following values:
[tex]\[ -3.0, -1.0, -0.42857142857142855, -0.14285714285714285, 0.14285714285714285, 0.42857142857142855, 1.0, 3.0 \][/tex]
These are the potential rational solutions for the given polynomial [tex]\( h(x) = -7x^3 + 4x^2 - 8x - 3 \)[/tex].
