expanding (x+y)^2 (from no algebra to abstract algebra)  Summary and Q&A
TL;DR
(x+y)^2 is not equal to x^2+y^2; instead, it expands to x^2+2xy+y^2, which is important to remember in abstract algebra.
Questions & Answers
Q: Why does (x+y)^2 not equal x^2+y^2?
The expansion of (x+y)^2 using the foil method shows that it is equal to x^2+2xy+y^2. The mistake in thinking it equals x^2+y^2 is due to the commutative property, which may not hold in abstract algebra.
Q: How can the area of a square help understand the expansion of (x+y)^2?
By visualizing (x+y) as the length and width of a square, the expansion can be understood as adding the areas: x^2 (area of a square with side x), 2xy (twice the area of a rectangle with sides x and y), and y^2 (area of a square with side y).
Q: What are the definitions of x and x^2 in calculus?
In calculus, x is defined as the integral of 1 from 0 to x. x^2 is defined as the integral of 2t from 0 to x. These definitions are used to explain the expansion of (x+y)^2 in terms of integrals.
Q: How is the expansion of (x+y)^2 derived using abstract algebra?
By using the definitions of x and x^2 as integrals, the expansion is written as the integral of 2t from 0 to x+y. Through substitution and properties of integrals, it simplifies to x^2+2xy+y^2, which is the correct expansion.
Summary & Key Takeaways

The content explains the correct expansion of (x+y)^2 as x^2+2xy+y^2.

It demonstrates the use of the foil method to expand the expression.

The video also introduces the concept of abstract algebra and how it challenges the traditional understanding of (x+y)^2.