Question

Subtract (x^3 - 5y^3 + 2z^3 + 3xyz) from sum of (-3x^3 - 7y^3 + z^3 + 5xyz), (3x^3 - y^3 + 7z^3 + 5xyz)and (-x^3 - y^3 - 2z^3 - 3xyz).

Answer 1

To solve the problem, we need to first find the sum of the three polynomials and then subtract the polynomial \( (x^3 - 5y^3 + 2z^3 + 3xyz) \) from this sum.

Let's denote the three polynomials as follows:

1. \( P_1 = -3x^3 - 7y^3 + z^3 + 5xyz \)
2. \( P_2 = 3x^3 - y^3 + 7z^3 + 5xyz \)
3. \( P_3 = -x^3 - y^3 - 2z^3 - 3xyz \)

First, we will sum these three polynomials.

**Step 1: Evaluating the sum \( P_1 + P_2 + P_3 \)**

Combine the coefficients of \( x^3 \), \( y^3 \), \( z^3 \), and \( xyz \):

- For \( x^3 \):
\[
-3 + 3 - 1 = -1
\]

- For \( y^3 \):
\[
-7 - 1 - 1 = -9
\]

- For \( z^3 \):
\[
1 + 7 - 2 = 6
\]

- For \( xyz \):
\[
5 + 5 - 3 = 7
\]

Thus, the sum is:
\[
P_1 + P_2 + P_3 = -1x^3 - 9y^3 + 6z^3 + 7xyz
\]

**Step 2: Now, let's subtract \( (x^3 - 5y^3 + 2z^3 + 3xyz) \) from the sum**

We will subtract the polynomial \( (x^3 - 5y^3 + 2z^3 + 3xyz) \) from the sum we calculated:

\[
(-1x^3 - 9y^3 + 6z^3 + 7xyz) - (x^3 - 5y^3 + 2z^3 + 3xyz)
\]

This operation will involve distributing the negative sign across the second polynomial:
\[
-1x^3 - 9y^3 + 6z^3 + 7xyz - x^3 + 5y^3 - 2z^3 - 3xyz
\]

Now, combine like terms:

- For \( x^3 \):
\[
-1 - 1 = -2
\]

- For \( y^3 \):
\[
-9 + 5 = -4
\]

- For \( z^3 \):
\[
6 - 2 = 4
\]

- For \( xyz \):
\[
7 - 3 = 4
\]

Thus, the resulting polynomial after the subtraction is:
\[
-2x^3 - 4y^3 + 4z^3 + 4xyz
\]

Finally, we can write down the result:
\[
\boxed{-2x^3 - 4y^3 + 4z^3 + 4xyz}
\]