Algebraic expressions

An algebraic expression is a combination of numbers, letters and symbols with a mathematical meaning. Examples: 3 - 2 ≠ 0; a + b = b + a; x + x = 2 · x; n³ = n · n · n; ax² + bx + c. As you can see there are many of the symbols that appear in the video.

Think of other symbols we use in mathematics.

Properties of the language of algebra: 

1. 🤔 It is not ambiguous.

2. 🌍 It is universal.

3. ⏳ It is concise.

Substitution

Substitution of a numerical value for the indeterminate of an algebraic expression consists in take out the variables, put instead one or more given numbers and make the operations. The result is a number. Example: substitute x by 3 in the following expression, x² + 1, gives you 10 (=3² + 1).

7. Substitution (valor numérico). Let's answer the questions at the bottom.

8. Substitute in the following algebraic expresions: a) x⁵ - x², when x = -1; b) a² + b², when a=1 and b=-1; c) 3n² - 5abc, when n=1, a=2, b=-1 and c=0; d) (3x²-5)/y, when x=5 and y=-14; e) 2b when b = -1; f) 1 - 2y when y = -2; g) -4bc² + 3b⁴ when b = 1 and c = -2; h) (x + y)² when x = 2 and y = 5; i) x³ - 5x² + 6x + 8 when x = -3 and when x = 3; j) 3x³y² - 5xy when x = -2 and y = -1

Monomials

A monomial is an algebraic expression that consists of a single term. An example of monomial is 5x⁴y³, where:

- a is a real number, called the coefficient.

- x and y are the variables raised to certain exponents.

Addition and Subtraction of Monomials

You can only add or subtract like monomials. For two monomials to be like, they must have the same type of variables with the same exponents.

-Like monomials:  4x² and -3x²

-Unlike monomials: 4x² and 3x.

Steps to add or subtract monomials:

1. Identify if the monomials are like. Compare the variables and their exponents.

2. Add or subtract the coefficients of like monomials, leaving the variables unchanged.

Example: 4x² + 3x² = (4 + 3) x² = 7x²

5x³ - 2x³ = (5 - 2) x³ = 3x³

If the monomials are not like, they cannot be added or subtracted.

Multiplication of Monomials

To multiply two monomials, you multiply the coefficients and apply the laws of exponents to the variables. So, follow these steps:

1. Multiply the coefficients of the monomials.

2. Apply the exponent rules: when multiplying variables with the same name, add their exponents.

Example: (3x²) · (4x³) = (3 · 4) · x²+³ = 12x⁵

If there are more than one variable, apply the same exponent addition rule to each variable.

Example: (2x²y) · (3xy²) = (2 · 3) · x²+¹ · y¹+² = 6x³y³

Division of Monomials

To divide monomials, divide the coefficients and subtract the exponents of the same variables. If the exponent in the numerator is greater than in the denominator, subtract. If the exponent is smaller, the result will be a negative exponent.

Example: 6x⁴/3x² = 6/3 · x⁴-² = 2x²

If there are more than one variable, apply the same exponent subtraction rule to each variable.

Example: 4x³y² / 2x²y = 4/2 · x³-² · y²-¹ = 2x¹y¹ = 2xy

Power of a Monomial

When raising a monomial to a power, raise both the coefficient and each variable to that power by multiplying the exponents.

Example: (3x²)³ = 3³ · x²·³ = 27x⁶

For more than one variable: (2x²y)³ = 2³ · x²·³ · y¹·³ = 8x⁶y³

Practice Activities. Perform the following operations: 

Easy level: a) 3x² + 5x² ; b) 7x³ - 2x³ ; c)  4x²y + 3x²y ; d) (2x) · (3x) ; e) (4x²) · (5x) ; f) (3xy) · (2x²y)

Intermediate Level: a) 6x⁴/3x² ; b) 8x³y/4xy ; c) 10x⁵/2x² ; d)  4x³ + 7x³ - 2x³ ; e) 6y² + 3y² - 8y² ; f) 5a²b - 3a²b + 2a²b 

Advanced Level: a) (3x²)³ ; b)  (2x³y)² ; c) (4xy²)⁴ ; d) (2x²) · (3x³y²) ; e) (5a³b²) · (2a²b) ; f) (x²y) · (4x³y³) 

Expert Level: a) (3x² + 5x²) · 2x ; b) (4x²y²)/(2xy) + 3x^2 ; c) (2a²b³) · (4a³b²)

Polynomials

Learn about polynomials in this website.

Repaso: el lenguaje algebraico.

Puedes practicar las operaciones con polinomios con esta ficha, que contiene las soluciones para que puedas comprobar si lo has hecho bien.

Pobres gatitos.

Answer key (Introductory video)

• The equal sign (=) was invented in the 16th century.  

• It was created by the mathematician Robert Recorde.  

• He chose this symbol (two parallel horizontal lines) because, according to him, nothing can be more equal than these two lines.  

• In fact, some people at one point used vertical line segments as the symbol for equality.  

• The plus sign (+) originated from condensing the Latin word "et," which means "and."  

• Other symbols mentioned in the video are: lines, dots, arrows, Latin letters, Greek letters, superscripts, subscripts, the exclamation mark (!) for factorials, the multiplication sign, exponents, and the uppercase sigma (∑).  

• The exclamation mark (!) represents the factorial and was introduced by Christian Kramp simply because he needed a shorthand for such expressions.  

• Characters represent unknown quantities, relationships between variables, and specific numbers that appear frequently but would be cumbersome or impossible to write out fully in decimal form. They can also represent entire groups of numbers and equations.  

• Some symbols are especially valuable as shortcuts because they condense repeated operations into a single expression. Two examples are:  

  1. The multiplication sign, which abbreviates repeated addition of the same number. For instance, instead of writing 3 + 3 + 3 + 3, we can write 3 × 4.  

  2. Exponents, which indicate that a number is multiplied by itself a certain number of times. For example, instead of writing 5 × 5 × 5, we can write 5³.