Multiple Choice
Find the product of the given complex number and its conjugate.
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0. Review of College Algebra
Complex Numbers
1:45 minutesProblem 15
Textbook Question
In Exercises 9–20, find each product and write the result in standard form.
(3 + 5i)(3 − 5i)
Verified step by step guidance1
Recognize that the expression is a product of two complex conjugates: \((3 + 5i)\) and \((3 - 5i)\).
Recall the formula for the product of conjugates: \((a + bi)(a - bi) = a^2 + b^2\), where \(a\) and \(b\) are real numbers.
Identify \(a = 3\) and \(b = 5\) from the given expression.
Calculate \(a^2\) and \(b^2\) separately: \$3^2\( and \)5^2$.
Add the results from the previous step to write the product in standard form: \(a^2 + b^2\).
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Video duration:
1mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Complex Numbers and Standard Form
Complex numbers are expressed in the form a + bi, where a is the real part and b is the imaginary part. Writing a complex number in standard form means expressing it explicitly as a sum of its real and imaginary components.
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Multiplication of Complex Numbers
To multiply complex numbers, use the distributive property (FOIL method), multiplying each term in the first complex number by each term in the second. Remember that i² equals -1, which simplifies the product.
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Difference of Squares Formula
The product (a + b)(a - b) equals a² - b². This formula applies to complex conjugates like (3 + 5i)(3 - 5i), simplifying the multiplication by turning it into a difference of squares involving real numbers and imaginary parts.
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