FUNDAMENTAL CONCEPTS OF ALGEBRA

FUNDAMENTAL CONCEPTS OF ALGEBRA

 

Arithmetic quantities are represented by numbers that represent given values.

 For example, the number 5 represents the value of five and if we want to express another quantity we must choose a number other than 5.

 Unlike Arithmetic, in Algebra quantities are studied in a general way, using quantities represented by symbols that are not only numbers but also others such as the letters of the alphabet that can represent any single value.

 A letter represents any value because that symbol will assume the value that we assign to it, and is unique. After all, within the same problem that letter cannot represent another value other than the one we have assigned.

 The range of problems that can be solved through algebra is extensive and to face a problem it is required to transform the situation into algebraic expressions containing symbols and signs.

 symbols

 Numbers

 They are used to represent known quantities and are determined by their value.

 

 They are used to represent all kinds of quantities, whether known or unknown.

 known quantities

 They are usually represented by the first letters of the alphabet (a, b, c, …).

 unknown quantities

 They are usually represented by the last letters of the alphabet (..., w, x, y, z).

 TIP

 The same letter can represent different quantities only if it is written differently, for example

 adding super indices (a'

 ≠ a'') or sub-indices (a1 ≠ a2 ).

 signs

 The signs are grouped into three classes:

 Operation signs:

 The operations are the same that we know from arithmetic, they and their respective symbols are:

 Addition or Sum: +

 Subtraction or Subtraction: –

 Multiplication or product: · or ×

 Division: ÷ or /

 Powers: a^n

where a is the base and n is the exponent of the power.

 roots: √a

 √a where a is the base and n is the exponent of the root, the sign √ is radical.

 relationship signs

 They are used to indicate the relationship between two quantities.

 =: Equality

 >: Strict major

 ≥: Greater than or equal to

 <: strict minor

 ≤: Less than or equal to 

Grouping signs

 Refers to the variety of parentheses ( ),[ ],{ }, among others, and are used to indicate that the operation between they must be done first.

 Now that we have defined the symbols and signs of algebra we can define an algebraic term and then an algebraic expression and their respective classifications.

 Algebraic term

 An algebraic term consists of a symbol or symbols that are not separated from each other by the signs of addition or subtraction.

 Example

a, -3b, 2xy^2,4a/3x.

 An algebraic term is made up of its sign, coefficient, its literal part, and its degree.

 - The sign of an algebraic term is positive or negative, when it is positive it is generally omitted to write

 The sign.

 - The coefficient of an algebraic term is the numerical quantity.

 - The literal part of an algebraic term corresponds to its letters including their exponents.

 - The degree of an algebraic term can be absolute (the sum of the exponents of all the factors of its

 literal part) or relative, that is, concerning each literal factor.

 Example:

 For the algebraic term 2xy^2

  determine its components:

 Positive sign

 Coefficient: 2

 Literal part: xy^2

 Absolute Degree: 1 + 2 = 3

 Degree relative to x: 1

 Degree relative to y: 2

 algebraic expression

 An algebraic expression is an algebraic term (monomial) or the addition or subtraction of two or more of them.

 (polynomial).

 Depending on the number of terms, polynomials receive some special names, so if a polynomial is made up of the addition or subtraction of two monomials, it is called a binomial, if it is made up of three terms added and/or subtracted is then called a trinomial.  If it is made up of the addition and/or subtraction of four or more terms is then generically called a polynomial.

Example:

 The absolute degree of a polynomial corresponds to the degree of its highest degree term, while the

 degree relative to a letter is the largest exponent of the letter in the polynomial.

 Example:

 Let be the polynomial ax4

 -5x3

 +x2

 -3x determine its absolute degree and relative to its literal factors

 Absolute grade: 5

 Degree relative to x: 4

 Degree relative to a: 1

 like terms

 Two or more terms are similar when they have the same literal part (including their exponents).

 Reduction of like terms

 The reduction of like terms is an operation whose objective is to convert them into a single term

 two or more like terms.

 Example:

 Recognize Like Terms:

 

 examples

 Reduce like terms

 a) 2y+5y=7y

 b) -2a+5a=3a

 c) ab+ ab+b = ab+b

 d) -m-3m-6m+5m=-5m

 e) 3a(x-2)+5a(x-2)-a(x-3)+2a(x-3)+yz2

 +yz=8a(x-2)+a(x-3)+yz2

 +yz

 SUGGESTED EXERCISES

 Reduce, if possible, the following polynomials:

 a) (a^2 -3a)-(3a + a^2)

 b) x^3 + y^2  - (3x^3  - 2 and 2 ) + (y^2  -x^3 ) - (4y^2- 6x^3)

Properties of the Royals

 From the study of the operations on the sets ℤ and ℚ, we conclude that subtractions can be

 consider as additions and divisions as multiplications and, therefore, + and · are the operations

 relevant to these sets.  Now, if we endow the set R with these same operations, we form an algebraic structure that, in this case, receives the name of Body.  This name indicates that the structure (ℝ, +, · ) fulfils several properties that allow us to use algebra as a tool effectively.

 TIP

 Considering the existence of negative numbers (set ℤ) we can represent a subtraction

 as an addition.

 Example

 5-7=-2 can be rewritten as 5 + (-7) =-28-4=4 can be rewritten as 8 + (-4)= 4

 TIP

 As for the division, this can be considered a multiplication thanks to the set of numbers

 Rational (ℚ) and also with the use of powers.

 Example

 

  It can be rewritten 5/2 or 5×2^-1.

 Next, we will symbolically indicate these properties:

 Addition properties in ℝ

 1. Closure: If a ∊ ℝ and b ∊ ℝ, then (a + b)∊ ℝ

 2. Associativity: If a, b, c ∊ ℝ, then a + (b + c) = (a + b) + c

 3. Commutativity: If a, b ∊ ℝ, then a + b = b + a

 4. Additive neutral element: If a ∊ ℝ, then there exists a unique additive neutral element, 0, such that to + 0 = to

 5. Additive or opposite inverse element.  For every element a ∊ ℝ, there exists an element -a ∊ ℝ, such that a + (-a) = 0

Properties of multiplication in ℝ.

 6. Closure: If a ∊ ℝ and b ∊ ℝ, then ab ∊ ℝ

 7. Associativity: If a, b, c ∊ ℝ, then a(bc) = (ab)c

 8. Commutativity: If a, b ∊ ℝ, then ab = ba

 9. Neutral multiplicative element: If a ∊ ℝ, then there exists a single neutral multiplicative element, 1, such that a 1 = a

 10. Inverse multiplicative or reciprocal element: For each element a-1 ∊ ℝ, except for 0, there is an element a · a-1 ∊ ℝ, such that.  The element a-1 is also often written

 For addition and multiplication in ℝ.

 11. Distributivity:If a, b, c ∊ ℝ, then a(b +c) = ab + ac

 Operation with polynomials

 Addition (and subtraction) of polynomials

 The addition and subtraction or addition and subtraction respectively of monomials has been exemplified in the reduction of similar terms and its operation is extensible to poly-monies as exemplified in the following

 solved exercises.

 examples be the following polynomials

 Polynomial 1: a^3b-b^4 +ab^3 +5a^2 +b^2

 Polynomial 2: -2a2b2+4ab3+2b4

 a) Add the polynomials:

 (a^3b-b^4 +ab^3+5a^2b^2)+(-2a^2b^2 +4ab^3+2b^4)

 =a^3b-b^4+ab^3 +5a^2b^2-2a^2b^2+4ab^3 +2b4

 =a^3b+5a^2b^2 -2a^2b^2 +4ab^3 +ab^3 +2b^4 -b^4

 =a^3b+3a^2b^2 +5ab^3 +b^4

 b) Subtract polynomial 2 from Polynomial 1:

 a^3b-b^4 +ab^3 +5a^2b^2 -(-2a^2b^2 +4ab^3 +2b^4)

 =a^3b-b^4+ab^3+5a^2b^2 +2a^2b^2-4ab^3 -2b^4

 =a^3b+5a^2b^2 +2a^2b^2 -4ab^3+ab^3 -2b^4 -b^4

 =a^3b+7a^2 b2 -3ab^3 -3b

Multiplication

 To find the product of two polynomials we use the distributive properties, the law of signs, the laws

 of the exponents and the reduction of like terms, as shown in the example below.

 Example

 be the following polynomials

 Polynomial 1: x³ +3x-1

 Polynomial 2: 2x^2 -4x+5

 We calculate the product by applying the distributive properties and then we make the product of

 monomials and the reduction of like terms expressing the final result, which is another polynomial, with

 its terms ordered from highest to lowest degree.

=(x^3+3x-1)(2x² -4x+5)

=(x3 +3x-1)(2x² )+(x^3+3x-1)(-4x)+(x^3+3x-1)(5)

=(x^3 )(2x^2)+(3x)(2x² )+(-1)(2x^2 )+(x^3 )(-4x)+(3x)(-4x)+(-1)(-4x )+(x^3)(5)+(3x)(5)+(-1)(5 )

=2x5+6x3 -2x2-4x4 -12x2 +4x+5x3+15x-5

=2x5-4x4+11x3 -14x2+19x-5

 

 Remarkable products

 Certain products of binomials occur so frequently that you must learn to recognize them, these

 are multiplications of easily recognizable algebraic expressions and to determine their development

 it is enough to apply a known general formula.

 binomial square

 Corresponds to the expression (x + y)^2

  or (x - y)^2

  which represents the product (x + y)(x + y) or (x - y)(x - y)

 respectively.

 Let us determine the general formula of its development.

 (x + y)^2

  = (x + y)(x + y)

 applying distributivity

 

  = (x + y)x + (x + y)y

 distributing again

 

  = xx + xy + xy + yy

 and reducing

 

  =x^2+ 2xy + y^2

 

 which is the sought formula.

 For the binomial (x - y)^2

 , its development formula is x^2

 - 2xy + y^2

 .

 Then, in general, we can write down:

(x±y)^2=x^2±2xy+y^2

examples

 a) (3a + 4)^2

 = (3a)^2+ 2•3a4 + 4^2

 = 9a^2+2•4a +16

 b) (a - 3b)^2

 = a^2 - 2-a-3b + (3b)^2

 = a^2- 6ab + 9b^2

Product of binomials with a common term

 It corresponds to the multiplication of two binomials where one of the terms is repeated in both.  It's way

 overall is:

 (x + a)(x + b)

 Let's determine the development formula using the same procedure as in the previous cases.

 (x + a)(x + b) = (x + a)x + (x + a)b

 = xx + ax + xb + ab