Question

Maliks work to find the difference of (1.8n^3p + 7n^2) - (8.4p^2 + 2n^3p - 1.9n^2) is shown.
Given: (1.8n^3p + 7n^2) - (8.4p^2 + 2n^3p - 1.9n^2)
Step 1: 1.8n^3p + 7n^2 - 8.4p^2 + 2n^3^p - 1.9n^2
Step 2: 1.8n^3 p + 2n^3p + 7n^2 - 1.9n^2 - 8.4p^2
Step 3: 3.8n^3p + 5.1n^2 - 8.4p^2
Complete the statements.
Malik ____ (choices are correctly found the difference of the polynomials, first made an error in Step 1, first made an error in Step 2, or first made an error in Step 3.)
The difference of (1.8n^3p + 7n^2) - (8.4p^2 + 2n^3p - 1.9n^2) is ____ (choices are -0.2n^3p + 8.9^n - 8.4p^2, -0.2n^3p + 5.1n^2 - 8.4p^2, 3.8n^3p + 8.9n^2 + 8.4p^2, or 3.8n^3p + 5.1n^2 - 8.4p^2.

Answer 1

Let's analyze Malik's steps to determine if there are any errors in finding the difference of the given polynomials.

The expression given is:
\[
(1.8n^3p + 7n^2) - (8.4p^2 + 2n^3p - 1.9n^2)
\]

We can rewrite this as:
\[
1.8n^3p + 7n^2 - 8.4p^2 - 2n^3p + 1.9n^2
\]

Now let's combine the like terms.

1. For the \(n^3p\) terms:
\[
1.8n^3p - 2n^3p = (1.8 - 2)n^3p = -0.2n^3p
\]

2. For the \(n^2\) terms:
\[
7n^2 + 1.9n^2 = (7 + 1.9)n^2 = 8.9n^2
\]

3. The \(p^2\) term remains:
\[
-8.4p^2
\]

Putting it all together, we get:
\[
-0.2n^3p + 8.9n^2 - 8.4p^2
\]

Now let's look at Malik's steps.

- **Step 1**: He wrote \(1.8n^3p + 7n^2 - 8.4p^2 + 2n^3^p - 1.9n^2\) and incorrectly included a "+" sign in front of \(2n^3p\) instead of a "−". That is an error.

- **Step 2**: He combined the like terms incorrectly. The valid combination of \(n^3p\) terms and \(n^2\) terms is not correctly stated in his work either.

Thus, Malik first made an error in **Step 1**.

Consequently, the correct difference of the polynomials is:
-0.2n^3p + 8.9n^2 - 8.4p^2.

So, to complete the statements:
Malik **first made an error in Step 1**.
The difference of \((1.8n^3p + 7n^2) - (8.4p^2 + 2n^3p - 1.9n^2)\) is **-0.2n^3p + 8.9n^2 - 8.4p^2**.