https://file.aiquickdraw.com/mathbot/file/1726425039281zs0wzccm.jpg

Let's begin by addressing the problem in the two parts presented:

### Part (a):
You need to evaluate the polynomial \( y = x^3 - 7x^2 + 8x - 0.35 \) at \( x = 1.37 \), using **3-digit arithmetic with chopping**. Then, the percent relative error should be computed.

### Part (b):
Next, you are asked to evaluate the polynomial by expressing it in the factored form \( y = ((x - 7)x + 8)x - 0.35 \) and also use **3-digit arithmetic with chopping**. You will then compare the error with part (a).

### Step 1: Evaluate Polynomial (a)

The polynomial is:
\[
y = x^3 - 7x^2 + 8x - 0.35
\]

We will compute the value of \( y \) at \( x = 1.37 \) using 3-digit arithmetic with chopping.

1. \( x^3 = (1.37)^3 = 2.573053 \), but with 3-digit chopping, this becomes **2.57**.
2. \( 7x^2 = 7(1.37)^2 = 7(1.8769) = 13.1383 \), but chopped to 3 digits: **13.1**.
3. \( 8x = 8(1.37) = 10.96 \), chopped to 3 digits: **10.9**.
4. Subtract \( 0.35 \).

Now, putting these values into the equation:
\[
y = 2.57 - 13.1 + 10.9 - 0.35 = 2.57 - 13.1 + 10.9 - 0.35
\]
After performing the operations step by step with 3-digit precision:
\[
y = -10.53 + 10.9 = 0.37
\]
Thus, the result from (a) is **0.37**.

### Step 2: Evaluate Polynomial (b)

Now express the polynomial in the factored form:
\[
y = ((x - 7)x + 8)x - 0.35
\]
Evaluate at \( x = 1.37 \) with 3-digit arithmetic and chopping:

1. First, \( x - 7 = 1.37 - 7 = -5.63 \), chopped to 3 digits: **-5.63**.
2. Multiply this by \( x \): \( (-5.63)(1.37) = -7.7181 \), chopped to 3 digits: **-7.71**.
3. Add 8: \( -7.71 + 8 = 0.29 \), chopped to 3 digits: **0.29**.
4. Multiply by \( x \): \( 0.29 \times 1.37 = 0.3973 \), chopped to 3 digits: **0.39**.
5. Subtract 0.35: \( 0.39 - 0.35 = 0.04 \).

Thus, the result from (b) is **0.04**.

### Step 3: Percent Relative Error

The exact value of \( y \) at \( x = 1.37 \) without chopping is approximately:

1. \( x^3 = 2.573053 \)
2. \( 7x^2 = 13.1383 \)
3. \( 8x = 10.96 \)

Thus, the exact value is:
\[
y = 2.573053 - 13.1383 + 10.96 - 0.35 = 0.044753
\]

Now calculate the percent relative error for both part (a) and (b):

1. **Part (a)**: 
   \[
   \text{Percent relative error} = \left| \frac{0.37 - 0.044753}{0.044753} \right| \times 100 = 726.66\%
   \]

2. **Part (b)**: 
   \[
   \text{Percent relative error} = \left| \frac{0.04 - 0.044753}{0.044753} \right| \times 100 = 10.62\%
   \]

### Conclusion:
- The value of \( y \) in part (a) is **0.37** with a relative error of **726.66%**.
- The value of \( y \) in part (b) is **0.04** with a relative error of **10.62%**.

Would you like further clarification or details on any specific part?

### 5 Related Questions:
1. How is 3-digit arithmetic with rounding different from chopping?
2. What is the importance of reducing relative error in numerical methods?
3. How can we minimize errors when performing arithmetic in floating-point calculations?
4. What is the exact difference between truncation and rounding errors?
5. Why does the factored form in part (b) result in lower relative error?

### Tip:
When working with numerical approximations, always consider how reformulating an expression might affect rounding and truncation errors!

Evaluate the polynomial y = x^3 - 7x^2 + 8x - 0.35 at x = 1.37 using 3-digit arithmetic with chopping. Then repeat the calculation by expressing y as ((x - 7)x + 8)x - 0.35 and compare the percent relative error.

Evaluate the polynomial y = x^3 - 7x^2 + 8x - 0.35 at x = 1.37 using 3-digit arithmetic with chopping. Compare the results with an alternative factored form and calculate the percent relative error. Suitable for students studying error analysis and polynomial evaluation in numerical methods.

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