Answer :
To find the standard form of [tex]\((7 - 5i)(2 + 3i)\)[/tex], we will use the distributive property (also known as the FOIL method for binomials) to multiply these complex numbers step-by-step:
1. First, multiply the real parts:
[tex]\[ 7 \times 2 = 14 \][/tex]
2. Outside, multiply the real part of the first complex number by the imaginary part of the second complex number:
[tex]\[ 7 \times 3i = 21i \][/tex]
3. Inside, multiply the imaginary part of the first complex number by the real part of the second complex number:
[tex]\[ -5i \times 2 = -10i \][/tex]
4. Last, multiply the imaginary parts:
[tex]\[ -5i \times 3i = -15i^2 \][/tex]
Note that [tex]\(i^2 = -1\)[/tex], so [tex]\(-15i^2\)[/tex] becomes:
[tex]\[ -15(-1) = 15 \][/tex]
Now, add together all these results:
[tex]\[ 14 + 21i - 10i + 15 \][/tex]
Combine the real parts and the imaginary parts:
[tex]\[ (14 + 15) + (21i - 10i) = 29 + 11i \][/tex]
Therefore, the standard form of [tex]\((7 - 5i)(2 + 3i)\)[/tex] is:
[tex]\[ \boxed{29 + 11i} \][/tex]
So the correct answer is:
C. [tex]\(29 + 11i\)[/tex]
1. First, multiply the real parts:
[tex]\[ 7 \times 2 = 14 \][/tex]
2. Outside, multiply the real part of the first complex number by the imaginary part of the second complex number:
[tex]\[ 7 \times 3i = 21i \][/tex]
3. Inside, multiply the imaginary part of the first complex number by the real part of the second complex number:
[tex]\[ -5i \times 2 = -10i \][/tex]
4. Last, multiply the imaginary parts:
[tex]\[ -5i \times 3i = -15i^2 \][/tex]
Note that [tex]\(i^2 = -1\)[/tex], so [tex]\(-15i^2\)[/tex] becomes:
[tex]\[ -15(-1) = 15 \][/tex]
Now, add together all these results:
[tex]\[ 14 + 21i - 10i + 15 \][/tex]
Combine the real parts and the imaginary parts:
[tex]\[ (14 + 15) + (21i - 10i) = 29 + 11i \][/tex]
Therefore, the standard form of [tex]\((7 - 5i)(2 + 3i)\)[/tex] is:
[tex]\[ \boxed{29 + 11i} \][/tex]
So the correct answer is:
C. [tex]\(29 + 11i\)[/tex]
