Comprehensive Study Guide for 1st Year Baccalaureate Mathematics: Question Bank and Theory

Administrative Background and Institutional Context

The provided material originates from the Republic of Ecuador, specifically the Unidad Educativa Presidente Tamayo. The institution is located on Calle Rocafuerte between Av. 9 de Octubre and Calle Quito, with the contact telephone number 800-019. Within the educational administrative structure, it is identified as part of Circuito C04 and carries the AMIE code 22H00064, situated in Francisco de Orellana (Fco. de Orellana). These study materials comprise a Question Bank (Banco de Preguntas) intended for the final assessment of students in their 3rd year of Baccalaureate (3ero de Bachillerato) for the 2025-2026 academic year. Although the target students are in their senior year, the content focuses on foundational mathematical principles classified under the 1st year of Baccalaureate (Primero de Bachillerato) curriculum, specifically within the domain of Numbers and Algebra ( N%5meros y %1lgebra).

Classification and Properties of Real Numbers

A fundamental aspect of algebra involves the correct classification of numbers within the real number system. In addressing the specific case of the number $-5$, various classifications must be evaluated for accuracy. The number $-5$ is not a natural number, as natural numbers typically include only positive integers ($1, 2, 3, ...$) and sometimes zero. Furthermore, it is not an irrational number because it can be expressed as a ratio of two integers. Crucially, the number $-5$ is classified correctly as a negative integer (entero negativo). It is not a positive integer (entero positivo), as indicated by the presence of the negative sign.

Fundamental Arithmetic: Multiples and Divisibility

Identifying multiples is a basic arithmetic skill necessary for higher-level algebraic manipulation. A multiple is the product of any quantity and an integer. For instance, in determining which number among the options $14$, $15$, $16$, and $19$ is a multiple of $3$, one must find which value is divisible by $3$ without a remainder. The number $15$ is the correct choice because it satisfies the equation $3 \times 5 = 15$. The other options ($14$, $16$, and $19$) do not result from the multiplication of $3$ by any integer, thus they are not multiples of $3$.

Calculation of the Greatest Common Divisor (MCD)

The Greatest Common Divisor (MCD - M%3aximo Com%3un Divisor) is the largest positive integer that divides each of the integers without a remainder. To find the MCD of $12$ and $18$, one can employ prime factorization or a division table. For the number $12$, the factors can be broken down as follows: $12 \div 2 = 6$, $6 \div 2 = 3$, and $3 \div 3 = 1$. This indicates that $12 = 2^2 \times 3$. For the number $18$, the factors are: $18 \div 2 = 9$, $9 \div 3 = 3$, and $3 \div 3 = 1$, which means $18 = 2 \times 3^2$. By identifying the common prime factors with the lowest exponents, we find $2^1 \times 3^1$. Therefore, the calculation is $2 \times 3 = 6$. The options provided for this task were $6$, $4$, $-7$, and $0$, with $6$ being the mathematically correct answer.

Exponential Operations and Repeated Multiplication

Exponentiation is an operation involving two numbers, the base and the exponent, which represents the repeated multiplication of the base. In the expression $2^5$, the base is $2$ and the exponent is $5$. This dictates that the number $2$ should be multiplied by itself five times. The step-by-step calculation is as follows: $2 \times 2 \times 2 \times 2 \times 2 = 32$. The evaluation of the options $16$, $18$, $32$, and $64$ confirms that $32$ is the accurate result of this power. Note that while the transcript contains a minor handwritten notation error (referencing $35$), the systematic multiplication correctly yields $32$.

Definitions and Identification of Monomials

In algebra, expressions are classified based on the number of terms they contain. A monomial is defined as an algebraic expression consisting of exactly one term. In the provided set of expressions—$3x + 5$, $2x - 7$, $3x$, and $6x - 9$—the only literal that matches the definition of a monomial is $3x$. The other expressions ($3x + 5$, $2x - 7$, and $6x - 9$) are binomials because they consist of two distinct terms separated by addition or subtraction operators. The transcript explicitly notes that $3x$ is selected because it has only one term (tiene un t%3ermino).

Analysis of Algebraic Expressions: Identifying Independent Terms

Algebraic expressions are often composed of variable terms and constant terms. An independent term (t%3ermino independiente) is a part of an algebraic expression that consists only of a number and does not contain a variable. In reviewing the expressions $3x - 1$, $2x + 8y$, $x^2 - 8x$, and $7x - 9z$, we look for the presence of a constant. In the expression $3x - 1$, the $-1$ is the independent term because its value does not depend on the variable $x$. In contrast, the expression $2x + 8y$ has two variable terms, $x^2 - 8x$ consists of terms with different powers of $x$, and $7x - 9z$ features two separate variables, meaning none of these contain an independent constant term.