AoPS: Lesson 5, Class Extensions¶

Source: AoPS Academy, Algebra 2, Fall Semester 2024¶

The problems below are listed roughly in order of difficulty. The earlier problems align with the class discussion, while the later ones extend the material and are more challenging. You may work on any problems of your choosing, collaborate with other students, and ask the teacher for assistance.

Problem 1: Compute the following¶

(a) \((1 + 2i)(1 - 2i)\)
(b) \((3 + i)(3 - i)\)
(c) \((2 + 3i)(2 - 3i)\)

What do you notice about these numbers?

Problem 2:¶

(a) Write 61 as a sum of two perfect squares.
(b) Use this to write 61 as \((a + bi)(a - bi)\), where \(a\) and \(b\) are integers.

Problem 3:¶

(a) Compute the product \((1 + 2i)(4 - i)\).
(b) Show that the product \((1 + 2i)(4 - i)(1 - 2i)(4 + i)\) is a sum of two perfect squares.

Problem 4:¶

Show that the product \((1 + 2i)(4 - i)(1 - 2i)(4 + i)\) is a product of two numbers, each of which is the sum of two perfect squares.

Problem 5:¶

(a) Write 13 as the sum of two perfect squares.
(b) Write 82 as the sum of two perfect squares.
(c) Write \(13 \times 82\) as the sum of two perfect squares.

Problem 6:¶

For integers \(a\), \(b\), \(c\), and \(d\), show that \((a^2 + b^2)(c^2 + d^2)\) is the sum of two perfect squares.

Problem 7:¶

(a) Find a way to write any square of a sum of squares, \((m^2 + n^2)^2\), with \(m > n\), as a sum of two positive squares.
(b) Find a right triangle with integer sides and a hypotenuse of length \(13 \times 82\).

This structure separates each part clearly while maintaining the order and clarity of the problems.