Математика с д-р Атанас Илчев

1 Problem Find the numerical value of the expression $a^2+b^2$ if $a+b=5$ and $ab=9$. Solution: Let's recall the formula for truncated multiplication $(a+b)^2=a^2+2ab+b^2$, which we have already reviewed here. Notice that $a^2+b^2=(a+b)^2-2ab$, this is because $(a+b)^2-2ab=a^2+2ab+b^2-2ab$, and the ones complement $2ab$ and $-2ab$ are truncated. Thus, we have already figured out how to represent $a^2+b^2$ as the sum and product of $a$ and $b$, hence $a^2+b^2=(a+b)^2-2ab=5^2-2.9=25-18=7.$ Problem 2 Prove the identity $(ab+cd)^2=(a^2+c^2)(b^2+d^2)-(ad-cb)^2.$ Solution: We need to prove that the left side of this equality is equal to the right side. We will simplify both sides of the equality and compare them, hence $LS=a^2b^2+2abcd+c^2d^2$. Now we simplify the right-hand side of the equality  $LS=a^2b^2+a^2d^2+b^2c^2+c^2d^2-(a^2d^2-2adcb+c^2b^2)=$ $=a^2b^2+a^2d^2+b^2c^2+c^2d^2-a^2d^2+2adcb-c^2b^2=a^2b^2+c^2d^2+2abcd$.  Thus we get that $LS=RS$ and the equality is an identity...

We continue with the next and last of the abbreviated multiplication formulas $(a\pm b)(a^2\mp ab+b^2)=a^3\pm b^3$. We will look at some problems to show some applications of it. Problem 1 Perform the multiplication $(3-x)(9+3x+x^2).$ Solution: Notice that given the expression $(3-x)(9+3x+x^2)$, we can write it in the form $(3-x)(3^2+3x+x^2).$ We will apply the formula $(a-b)(a^2+ab+b^2)$, replacing $a$ with $3$ and $b$ with $x$, so we get $(3-x)(3^2+3x+x^2)=3^3-x^3.$ Problem 2 Perform the multiplication $(3t+2)(9t^2-6t+4).$ Solution: Given an expression, we can write it in the form $(3t+2)[(3t)^2-3t.2+2^2]$. Now it is easy to see that we can apply the formula $(a+b)(a^2-ab+b^2)$, where $a=3t$ and $b=2$, so we get $(3t+2)[(3t)^2-3t.2+2^2]=(3t)^3+2^3=27t^3+8.$ Problem 3 Simplify the expression $(x-2)(x^2+2x+4)-x(x-2)(x+2)-4(x-2).$ Solution: Apply the formulas $(a-b)(a^2+ab+b^2)=a^3-b^3$ and $(a-b)(a+b)=a^2-b^2$, hence $(x-2)(x^2+2x+4)-x(x-2)(x+2)-4(x-2)=x^3-2^3-x(x^2-4)-4x+8=x^3-8...

We continue with the next of the abbreviated multiplication formulas $(a\pm b)^3=a^3\pm 3a^2b+3ab^2\pm b^3$. Let's look at some problems to illustrate some of its applications. Problem 1 Perform the grading $(x+2)^3$. Solution: To solve this problem we will apply the formula $(a+b)^3=a^3+3a^2b+3ab^2+b^3$, where in our case $a=x$ and $b=2$, so we get $(x+2)^3=x^3+3x^2.2+3x.2^2+2^3=x^3+6x^2+12x+8$. Problem 2 Perform the $(2m-3n)^3$ grading. Solution:  $(2m)^2.3n+3.(2m).(3n)^2-(3n)^3=8m^3-36m^2n+54mn^2-27n^3.$ 3 Problem Perform the grading $(3t+z^2)^3$ Solution: To solve this problem we will apply the formula $(a+b)^3=a^3+3a^2b+3ab^2+b^3$, where $a=3t$ and $b=z^2$, hence $(3t+z^2)^3=(3t)^3+3. (3t)^2.z^2+3.(3t)(z^2)^2+(z^2)^3=27t^3+27t^2z^2+9z^4t+z^6.$ 4 Problem Simplify the expression $(x+1)^3-2(x-1)^2.$ Solution: Notice that the expression we need to simplify involves two of the shortcut multiplication formulas $(a+b)^3=a^3+3a^2b+3ab^2+b^3$ and $(a-b)^2=a^2-2ab+b^2$, the latter...

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...

With this and the following lessons, we will try together to overcome the difficulties in solving various problems in which these formulas are applied. Let us recall them before we begin: 1. $(a \pm b)^{2}=a^{2}\pm 2ab+b^{2}$; 2. $(a-b)(a+b)=a^{2}-b^{2}$; 3. $(a \pm b)^{3}=a^{3}\pm 3a^{2}b+3ab^{2}\pm b^{3}$; 4. $(a \pm b)(a^{2}\pm ab+b^{2})=a^{3}\pm b^{3}$. In this article, we consider the formulas $(a\pm b)^{2}=a^{2}\pm 2ab+b^{2}$. Let's solve a few problems to show how we will apply them: Problem 1 Perform the grading $(2x+y)^{2}.$ Solution: Let's consider the formula $(a+b)^{2}=a^{2}+2ab+b^{2}$. In our expression, $2x$ plays the role of $a$ and $y$ plays the role of $b$. Let us now write $2x$ instead of $a$ and $y$ instead of $b$. Thus we get that $(2x+y)^{2}=(2x)^{2}+2.2x.y+y^{2}$. Now we need to exponentiate $(2x)^{2}$. Let's recall the following property learned in 6th grade $(a.b)^{n}=a^{n}.b^{n}$, hence $(2x)^{2}=2^{2}.x^{2}=4x^{2}$. Let us now, having made this c...

Машината на Рамануджан – когато алгоритъмът открива математика | Д-р Атанас Илчев 📞 Онлайн уроци по математика за цялата страна ◆ гл.ас. д-р Атанас Илчев ◆ Индивидуални и групови уроци • Тел: 0883 375 433 ◆ Подготовка за НВО, ДЗИ, кандидатстудентски изпити ◆ 📞 Онлайн уроци по математика за цялата страна ◆ гл.ас. д-р Атанас Илчев ◆ Индивидуални и групови уроци • Тел: 0883 375 433 ◆ Подготовка за НВО, ДЗИ, кандидатстудентски изпити ◆ ★ Интересно от математиката Машината на Рамануджан — когато алгоритъмът открива математика Какво би се случило, ако компютър можеше да имитира математическата интуиция на един гений? Изследователи от Technion са изградили точно такава система — алгоритъм, кръстен на Сриниваса Рамануджан, който самостоятелно генерира нови математически...

Простите числа на Мерсен и GIMPS – търсенето на най-голямото просто число | Д-р Атанас Илчев 📞 Онлайн уроци по математика за цялата страна ◆ гл.ас. д-р Атанас Илчев ◆ Индивидуални и групови уроци • Тел: 0883 375 433 ◆ Подготовка за НВО, ДЗИ, кандидатстудентски изпити ◆ 📞 Онлайн уроци по математика за цялата страна ◆ гл.ас. д-р Атанас Илчев ◆ Индивидуални и групови уроци • Тел: 0883 375 433 ◆ Подготовка за НВО, ДЗИ, кандидатстудентски изпити ◆ ★ Интересно от математиката Простите числа на Мерсен — търсенето на най-голямото просто число Числата от вида \(2^p - 1\) крият тайна от хиляди години. Хиляди доброволци по целия свят дарят изчислителната мощ на компютрите си, за да намерят следващото рекордно просто число. Защо? И какво прави тези числа толкова специални? ...