Find each product and write the result in standard form. (7 - 5i)(- 2 - 3i)

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Recall that to multiply two complex numbers, you use the distributive property (also known as FOIL for binomials): multiply each term in the first complex number by each term in the second complex number.

Write the expression explicitly: $(7 - 5i)(-2 - 3i) = 7 \times (-2) + 7 \times (-3i) + (-5i) \times (-2) + (-5i) \times (-3i)$.

Calculate each product separately: $7 \times (-2) = -14$, $7 \times (-3i) = -21i$, $(-5i) \times (-2) = 10i$, and $(-5i) \times (-3i) = 15i^2$.

Remember that $i^2 = -1$, so replace \$15i^2$ with $15 \times (-1) = -15$.

Combine the real parts $(-14)$ and $(-15)$, and combine the imaginary parts $(-21i)$ and \$10i$ to write the product in standard form $a + bi$.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Complex Numbers

Complex numbers are numbers in the form a + bi, where a and b are real numbers and i is the imaginary unit with the property i² = -1. They extend the real number system and are used to represent quantities involving the square root of negative numbers.

Multiplication of Complex Numbers

To multiply complex numbers, use the distributive property (FOIL method) to expand the product, then combine like terms. Remember to replace i² with -1 to simplify the expression into standard form a + bi.

Standard Form of a Complex Number

The standard form of a complex number is a + bi, where a is the real part and b is the imaginary part. Writing the product in this form means expressing the result as a sum of a real number and an imaginary number.

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