Evaluate g(x) = x³ - 6.1x²+3.2x + 1.5 at x = 4.71 using three-digit chopping arithmetic and three digit rounding arithmetic. Hence compute the absolute and relative errors.

To evaluate \( g(x) = x^3 - 6.1x^2 + 3.2x + 1.5 \) at \( x = 4.71 \) using both three-digit chopping arithmetic and three-digit rounding arithmetic, follow these steps:

### Three-Digit Chopping Arithmetic
Chopping means truncating the numbers to three digits without rounding.

1. Compute \( x^3 \):
   \[
   4.71^3 = 104.787111 \approx 104 \text{ (chopped to three digits)}
   \]

2. Compute \( 6.1x^2 \):
   \[
   6.1 \times (4.71^2) = 6.1 \times 22.1841 = 135.32441 \approx 135 \text{ (chopped to three digits)}
   \]

3. Compute \( 3.2x \):
   \[
   3.2 \times 4.71 = 15.072 \approx 15.0 \text{ (chopped to three digits)}
   \]

4. The constant term remains unchanged.

Using these chopped values, we have:
\[
g(4.71) \approx 104 - 135 + 15.0 + 1.5 = -14.5
\]

### Three-Digit Rounding Arithmetic
Rounding means adjusting the numbers to the nearest three-digit number.

1. Compute \( x^3 \):
   \[
   4.71^3 = 104.787111 \approx 105 \text{ (rounded to three digits)}
   \]

2. Compute \( 6.1x^2 \):
   \[
   6.1 \times (4.71^2) = 6.1 \times 22.1841 = 135.32441 \approx 135 \text{ (rounded to three digits)}
   \]

3. Compute \( 3.2x \):
   \[
   3.2 \times 4.71 = 15.072 \approx 15.1 \text{ (rounded to three digits)}
   \]

4. The constant term remains unchanged.

Using these rounded values, we have:
\[
g(4.71) \approx 105 - 135 + 15.1 + 1.5 = -13.4
\]

### Exact Calculation
Now, calculate the exact value of \( g(4.71) \):
\[
g(4.71) = 4.71^3 - 6.1 \times 4.71^2 + 3.2 \times 4.71 + 1.5
\]
\[
= 104.787111 - 135.32441 + 15.072 + 1.5
\]
\[
= -14.965299
\]

### Absolute and Relative Errors
1. **Chopping Arithmetic:**

   - Absolute Error:
     \[
     |g_{\text{exact}} - g_{\text{chop}}| = |-14.965299 - (-14.5)| = | -14.965299 + 14.5| = 0.465299
     \]

   - Relative Error:
     \[
     \left|\frac{g_{\text{exact}} - g_{\text{chop}}}{g_{\text{exact}}}\right| = \left|\frac{-14.965299 - (-14.5)}{-14.965299}\right| = \left|\frac{0.465299}{-14.965299}\right| \approx 0.0311
     \]

2. **Rounding Arithmetic:**

   - Absolute Error:
     \[
     |g_{\text{exact}} - g_{\text{round}}| = |-14.965299 - (-13.4)| = | -14.965299 + 13.4| = 1.565299
     \]

   - Relative Error:
     \[
     \left|\frac{g_{\text{exact}} - g_{\text{round}}}{g_{\text{exact}}}\right| = \left|\frac{-14.965299 - (-13.4)}{-14.965299}\right| = \left|\frac{1.565299}{-14.

Learn how to evaluate the polynomial g(x) = x³ - 6.1x² + 3.2x + 1.5 at x = 4.71 using three-digit chopping and rounding arithmetic methods. Calculate absolute and relative errors for both methods.

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