To find the standard form of the product of the complex numbers [tex]\((7 - 5i)(2 + 3i)\)[/tex], we can use the distributive property (often referred to as FOIL when working with binomials). Here are the steps to follow:
1. Multiply the real parts together:
[tex]\[ 7 \times 2 = 14 \][/tex]
2. 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. 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. Multiply the imaginary parts together:
[tex]\[ (-5i) \times 3i = -15i^2 \][/tex]
Since [tex]\(i^2 = -1\)[/tex], we can simplify [tex]\(-15i^2\)[/tex]:
[tex]\[ -15i^2 = -15(-1) = 15 \][/tex]
5. Combine all the real parts and all the imaginary parts:
[tex]\[ 14 + 15 = 29 \][/tex]
[tex]\[ 21i - 10i = 11i \][/tex]
Thus, the standard form of [tex]\((7 - 5i)(2 + 3i)\)[/tex] is:
[tex]\[ 29 + 11i \][/tex]
Therefore, the correct answer is:
A. [tex]\(29 + 11i\)[/tex]
