To identify and explain Donte's mistake, let's simplify the expression [tex]\(4(1 + 3i) - (8 - 5i)\)[/tex] step by step correctly:
1. Applying the distributive property:
- For [tex]\(4(1 + 3i)\)[/tex]:
[tex]\[
4 \cdot 1 + 4 \cdot 3i = 4 + 12i
\][/tex]
2. Distributing the minus sign:
- For [tex]\(-(8 - 5i)\)[/tex]:
[tex]\[
-8 + 5i
\][/tex]
3. Combining the results:
- Add [tex]\(4 + 12i\)[/tex] and [tex]\(-8 + 5i\)[/tex]:
[tex]\[
(4 + 12i) - (8 - 5i) = 4 + 12i - 8 + 5i
\][/tex]
4. Simplifying the final expression:
- Combine the real numbers: [tex]\(4 - 8 = -4\)[/tex]
- Combine the imaginary parts: [tex]\(12i + 5i = 17i\)[/tex]
[tex]\[
-4 + 17i
\][/tex]
Now, let's look at Donte's steps and identify the mistake:
1. Donte rewrote the expression as:
[tex]\[
4 + 3i - 8 + 5i
\][/tex]
This step is incorrect because it seems like Donte mistakenly added the real number 4 and the imaginary coefficient [tex]\(3i\)[/tex] individually without correctly applying the distributive property.
Thus, Donte's mistake was:
He added the real number and coefficient of [tex]\(4(1+3i)\)[/tex] instead of multiplying.
