Find the imaginary part of the complex number if `z=( 2-i)(5+i)`

Answer

To find the imaginary part of the complex number \( z = (2 - i)(5 + i) \), we will follow these steps:

### Step 1: Expand the expression
We start with the expression:
\[ z = (2 - i)(5 + i) \]
Using the distributive property (also known as the FOIL method for binomials), we can expand this:
\[ z = 2 \cdot 5 + 2 \cdot i - i \cdot 5 - i \cdot i \]

### Step 2: Calculate each term
Now, we calculate each term:
- \( 2 \cdot 5 = 10 \)
- \( 2 \cdot i = 2i \)
- \( -i \cdot 5 = -5i \)
- \( -i \cdot i = -i^2 \) and since \( i^2 = -1 \), this becomes \( -(-1) = 1 \)

Putting it all together, we have:
\[ z = 10 + 2i - 5i + 1 \]

### Step 3: Combine like terms
Now, we combine the real and imaginary parts:
\[ z = (10 + 1) + (2i - 5i) = 11 - 3i \]

### Step 4: Identify the imaginary part
In the standard form of a complex number \( z = a + bi \), the imaginary part is the coefficient of \( i \). Here, we have:
\[ z = 11 - 3i \]
Thus, the imaginary part of \( z \) is:
\[ -3 \]

### Final Answer
The imaginary part of the complex number \( z \) is \( -3 \).

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