Математика с д-р Атанас Илчев

We continue with the next of the abbreviated multiplication formulas $(a-b)(a+b)=a^2-b^2$. Let's look at some problems to illustrate its applications. Problem 1 Perform the multiplication $(x+y)(x-y)$. Solution: Now we apply the formula $(a-b)(a+b)=a^2-b^2$, where $a=x$ and $b=y$, hence $(x-y)(x+y)=x^2-y^2$.  Problem 2 Perform the multiplication $(3x-4y)(3x+4y)$. Solution: Apply the formula $(a-b)(a+b)=a^2-b^2$, where $a=3x$ and $b=4y$, hence $(3x-4y)(3x+4y)=(3x)^2-(4y)^2=9x^2-16y^2$. 3 Problem Perform the multiplication $(x^2-z)(x^2+z)$. Solution: Apply the formula $(a-b)(a+b)=a^2-b^2$, where $a=x^2$ and $b=z$, hence $(x^2-z)(x^2+z)=(x^2)^2-(z)^2=x^4-z^2$. Let us recall the power grading property, i.e. $(a^n)^m=a^{n.m}$. 4 Problem Calculate $17.23$ in a rational way. Solution: Represent the product $17.23$ in the following way $17.23=(20-3)(20+3)$ and apply the formula $(a-b)(a+b)=a^2-b^2$, so $17.23=(20-3)(20+3)=20^2-3^2=400-9=391$. 5 Problem Simplify the expression $(3x-...

We will give definitions of some of the basic concepts, which we will explain with concrete examples. Definition 1: A rational expression that has no variables in the denominator is called an integer rational expression. Definition 2:  Monomial, we will call an integer rational expression, which is a product of letters and numbers. One's are also any number, variable or parameter. Definition 3: We will say that a monomial is in normal form when it is written with only one numerical multiplier, which stands in first place and is called a quotient, and any product of ones is written as a power. Example: Let's consider the monomial $3x.x.y.4.y.z$. Obviously, this monomial is not in normal form because in its notation we have two numerical factors $3$ and $4$, and also, products of equal letters that are not written as a power. The normal form of the monomial would be $3x.x.y.4.y.z=3.4.x.x.y.y.z=12x^2y^2z.$ As you can see, $12x^2y^2z$ is obviously in normal form because it satisf...