Which of the following is equivalent to the expression [tex]$(2i + 1)(5 - i)$[/tex]?

A. [tex]3 + 9i[/tex]
B. [tex]3 - 9i[/tex]
C. [tex]7 + 9i[/tex]
D. [tex]7 - 9i[/tex]



Answer :

Let's solve the expression [tex]\((2i + 1)(5 - i)\)[/tex] step-by-step and determine which of the given options it is equivalent to.

First, expand the expression using the distributive property (also known as the FOIL method for binomials):

[tex]\[ (2i + 1)(5 - i) = (2i \cdot 5) + (2i \cdot -i) + (1 \cdot 5) + (1 \cdot -i) \][/tex]

Now, solve each term individually:

1. [tex]\(2i \cdot 5 = 10i\)[/tex]
2. [tex]\(2i \cdot -i = -2i^2 = 3. \(1 \cdot 5 = 5\)[/tex]
4. [tex]\(1 \cdot -i = -i\)[/tex]

Next, note that [tex]\(i^2 = -1\)[/tex]. Therefore, the term [tex]\(-2i^2\)[/tex] simplifies as follows:
[tex]\[ -2i^2 = -2(-1) = 2 \][/tex]

Combining all the simplified terms:

[tex]\[ 10i + 2 + 5 - i \][/tex]

Group the real parts and the imaginary parts:

[tex]\[ (2 + 5) + (10i - i) \][/tex]

Simplify these grouped terms:

[tex]\[ 7 + 9i \][/tex]

Thus, the expression [tex]\((2i + 1)(5 - i)\)[/tex] simplifies to [tex]\(7 + 9i\)[/tex].

So, the equivalent expression is option [tex]\( C \)[/tex]:

C. [tex]\(7 + 9i\)[/tex]