Algebra 2nd ESO PDF

# Con lo que ya sabes, resuelve Suppose we build triangular structures using sticks and modeling clay, as shown in the illustration. We wonder how many clay balls and sticks we need depending on the size of the construction. Would you be able to complete the empty boxes in this table? Do it in y...

# Con lo que ya sabes, resuelve Suppose we build triangular structures using sticks and modeling clay, as shown in the illustration. We wonder how many clay balls and sticks we need depending on the size of the construction. Would you be able to complete the empty boxes in this table? Do it in your notebook. | Number of floors | Number of clay balls | Number of sticks | |---|---|---| | 1 | 3 | 3 | | 2 | 6 | 9 | | 3 | 10 | 18 | | 4 | 15 | 30 | | 5 | 21 | 45 | | 6 | 28 | 63 | On the other hand, we can also count how many cells (triangles with) appear in each case: | Number of floors | Number of triangles | |---|---| | 1 | 1 | | 2 | 4 | | 3 | 10 | | 4 | 20 | | 5 | 35 | | 6 | 56 | Finally, based on the above, which of the expressions you see below, on the left, serve to calculate the number of clay balls, B, the number of sticks, P, and the number of triangular cells, C, calling "n" the number of floors? Copy the table on the right into your notebook and place them in their place: | Floors | Clay balls | Sticks | Cells | |---|---|---|---| | n | $B=$... | $P=$... | $C=$... | ## El álgebra: ¿para qué sirve? Algebra covers the part of mathematics where letters are used to express numbers of unknown, variable or indeterminate value. It constitutes a language that facilitates the construction and description of mathematical processes. Let's look at some examples of the applications of algebra. ### Expressing arithmetic properties - The sum is associative, but the subtraction is not. $(a+b)+c=a+(b+c)$ $(a-b)-ca-(b-c)$ - Multiplication is distributive with respect to addition. $a. (b+c)=ab+a.c$ ### Generalizing numerical series (General term) | Example | 4 | 1 | a2 | 4 | 3 | a4 | a5 | |---|---|---|---|---|---|---|---| | ↓ | ↓ | ↓ | ↓ | ↓ | ↓ | ↓ | ↓ | | a = | 0 | 2 | 6 | 12 | 20 | (n-1).n | | | (0.1) | (1.2) | (2.3) | (3.4) | (4.5) | | | So, for example, if we want to know the tenth term in the series: 109.1090 or else 410 102-10 = 90 ### Expressing relationships between magnitudes (Formulas) - The area, A, of a rhombus of diagonals d1 and d2 → $A=\frac{d_{1}d_{2}}{2}$ - The interest, I, produced in months by a capital, C, placed in the bank at r% per year. → $I = \frac{C·r·n}{12·100}$ ### Expressing and operating numbers of indeterminate value (Algebraic expressions) **Examples:** - A natural number → a - The following → a+1 - Double the following →2. (a+1) - The square of the following → $(a+1)^2$ ### Expressing relationships that facilitate the resolution of problems (Equations) **Example:** In a test of 20 questions, 3 points are earned for each correct answer and 2 are lost for each incorrect or blank answer. How many correct answers and how many mistakes did Begoña make if she got 40 points? - Correct answers → x - Wrong answers → 20-x | POINTS | POINTS | TOTAL SCORE | |---|---|---| | EARNED | LOST | OBTAINED | | 3x | 2.(20 - x) | 40 | Verification:3.16-2(20-16) = 40 3x-2(20-x) = 40 x = 16 Solution: Begoña made 16 correct answers and 4 mistakes. ## Monomios - A monomial is the product of a known value (coefficient) by one or more unknown values, represented by letters (literal part). - **Coefficient:** The numerical part of a monomial - **Literal part:** The part of the monomial made up of letters raised to powers. - **Degree:** The sum of the exponents of all letter variables in the monomial, or if no variables are present, then the degree of the polynomial is 0. **EXAMPLE:** - **Monomial:** -5ax² - **Coefficient:** -5 \- **Literal part:** ax² - **Degree:** 3 **EXAMPLE:** - **Monomial:** 5x² - **Coefficient:** 5 \- **Literal part:** x² - **Degree:** 2 **EXAMPLE:** - **Monomial:** 3xy² - **Coefficient:** 3 \- **Literal part:** xy² - **Degree:** 3 **EXAMPLE:** - **Monomial:** 4a² \- **Coefficient:** 4 - **Literal part:** a² - **Degree:** 2 **EXAMPLE:** - **Monomial:** 5x²y² \- **Coefficient:** 5 - **Literal part:** x²y² - **Degree:** 4 **Degree of a monomial:** The degree of a monomial is the number of factors in the literal part. - $4a^2$: Second degree monomial - $5x^2y^2$ → x.x.yy: Fourth degree monomial **Numerical value of a monomial:** The numerical value of a monomial is the value of the monomial when the letters take specific values. - The numerical value of $2ab^2$ for $a = 1$ and $b = 2$ is $8$. - $2ab^2 = 2 * 1 * 2^2 = 8$ **Similar Monomials:** Two monomials are said to be similar when they have the same literal part. - $3a$ and $-2a$ ARE similar. - $4x^2y$ and $x^2y$ ARE similar. - $5x^2$ and $x^2$ ARE similar. **Sum of Monomials:** - Two monomials can only be added if they are similar. In that case, the coefficients are added, leaving the same literal part. - If the monomials are not similar, the sum is indicated. **Examples:** - $5a+2a=7a$ - $8x^2 3x^2 = 5x^2$ - $3x + 2x^2$ → remains indicated - $a^2 - a+ a^2- 2a^2 - a$ remains indicated **Expressions algebraicas:** - An expression made up of letters and numbers is called an algebraic expression. - **Binomials:** The sum (or subtraction) indicated of two monomials. - **Trinomials:** The sum (or subtraction) indicated of three monomials. - **Polynomials:** In general, the sum (or subtraction) of several monomials. **Example:** | POLYNOMIO | |---|---| | $a^2-1$ | BINOMIOS | | $x + y$ | | | $x²-3x+1$ | | | $a²-ab+2$ | | | $5x^4 - 3x^3 + 2x - 1$ | TRINOMIOS | | $x^5-4x^3 + 1$ | POLINOMIOS | **Degree of a polynomial:** The degree of a polynomial is the highest of the degrees of the monomials that make it up. **Example:** | POLYNOMIO | DEGREES | GRADO | |---|---|---| | $8x^3 -5x^2 + 7$ | 3, 2, 0 | 3 | | $x^5 - 4x^3 + 1$ | 5, 3, 0 | 5 | | $3x^2 + 11x$ | 2, 1 | 2 | **Numerical value of a polynomial:** When letters take specific values in a polynomial, the polynomial also takes a specific value. **Example:** Given the polynomial $3x^2 - 2x + 5$: - For x = 0: $3 * 0^2 - 2 * 0 + 5 = 0 - 0 + 5 = 5$ - For x = -2: $3 * (-2)^2 - 2 * (-2) + 5 = 12 + 4 + 5 = 21$ The numerical value of $3x^2-2x+5$ for *x*= 0 is 5. The numerical value of $3x^2-2x+5$ for *x*= -2 is 21. The numerical value of a polynomial depends on the value that the letters take. **Opposite of a polynomial:** The opposite of a polynomial is another polynomial with the same monomials changed in sign, plus for minus and minus for plus. **Example:** | POLYNOMIOS | OPUESTOS | |---|---| | $7x^3 + 3x^2 - 5x + 8$ | $-7x^3 + 3x^2 + 5x - 8$ | **Multiplication of monomials:** When multiplying two monomials, remember that each monomial is a product of numbers and letters, resulting in the product of two monomials being another monomial. **Example:** - (3a) (2a) =(3.2.a.a)=6a^2 - (5x) (-3x²) = 5 (-3)xx2 = -15x^3 - (3a). (ab) = 3.5.a.a. b = a^2b = 2a^2b **Division of monomials:** The quotient of two monomials can be a number, another monomial or an algebraic fraction. **Examples:** - 10a^2 : 5a^2 = - 10 * a * a ------------------ 5 * a * a = 2 - 18x^3 : 3x^2 = - 6 * 3 * x * x * x ------------------ 3 * x * x = 6x - (2a^2) : (6ab) = - 2 * a * a ------------------- 2 * 3 * a * b = a / (3b) Keeping in mind that letters represent numbers, in operations with algebraic expressions, all the properties of numerical operations are maintained. **Product of a polynomial by a monomial:** To multiply a polynomial by a monomial, multiply the monomial by each of the terms of the polynomial (distributive property). Observe how we multiply the polynomial x^3 - 4x^2 + 5x - 1 first by a number (2), then by a first-degree monomial (-3x) and finally by a second-degree monomial (x^2). | | x^3 - 4x^2 + 5x - 1 | 2 | x^3 - 4x^2 + 5x - 1 | -3x | x^3 - 4x^2 + 5x - 1 | x^2 | |---|---|---|---|---|---|---| | | x^3 - 4x^2 + 5x - 1 | x^3 - 4x^2 + 5x - 1 | x^3 - 4x^2 + 5x - 1 | | | | x^3 - 4x^2 + 5x - 1 | x^3 - 4x^2 + 5x - 1 | x^3 - 4x^2 + 5x - 1 | | | * | | * | * | * | * | * | | * * | | * | * | * | * | * * | | | 2x^3 - 8x^2 + 10x - 2 | | -3x^4 + 12x^3 - 15x^2 + 3x | | x^5 - 4x^4 + 5x^3 - x^2 | **Product of two polynomials**: Combining previous products, we can calculate the product of two polynomials by following a procedure similar to the one you use when multiplying numbers. | | x^3 - 4x^2 + 5x - 1 | x^2 - 3x + 2 | |---|---|---| | | x^3 - 4x^2 + 5x - 1 | x^2 - 3x + 2 | | | x^3 - 4x^2 + 5x - 1 | x^2 - 3x + 2 | | * | | * | | * * | | * | | | 2x^3 - 8x^2 + 10x - 2 | | | -3x^4 + 12x^3 - 15x^2 + 3x | | | x^5 - 4x^4 + 5x^3 - x^2 | - $(x^3-4x^2+5x-1).(x^2-3x+2) = $ - $ = (x^3-4x^2+5x-1).x^2 +$ - $ + (x^3-4x^2+5x-1).(-3x) +$ - $ + (x^3-4x^2+5x-1).2$ - $(x^3-4x^2+5x-1)(x^2-3x+2) = x^5 - 7x^4 + 19x^3 - 24x^2 + 13x - 2$ ## Productos notables - We call products notable for certain products of binomials whose memorization is useful to shorten some calculations with algebraic expressions. ### Square of a sum To begin, observe: - $(x+1)^2$ → $ x + 1$ | * | x + 1 | |---|---| | x | x^2 + x | | 1 | x + 1 | | | x^2 + 2x + 1 | - $(x + 5)^2$ → $x + 5$ | * | x + 5 | |---|---| | x | x^2 + 5x | | 5 | 5x + 25 | | | x^2 + 10x + 25 | - $(3x + 2)^2$ → $3x + 2$ | * | 3x + 2 | |---|---| | 3x | 9x^2 + 6x | | 2 | 6x + 4 | | | 9x^2 + 12x + 4 | The square of the sum of two monomials is equal to: - the square of the first, - plus twice the product of the first by the second, - plus the square of the second. $(a+b)^2= a^2 + 2ab + b^2$ Example: - $(3x+4)^2 = (3x)^2 + 2 * 3x * 4 + 4^2 = 9x^2 + 24x + 16$ ### Square of a diference In the same way: - $(x+3)^2=x^2+2*x*3+3^2 = x^2+6x+9$ - $(x-5)^2=x^2-2*x*5+5^2 = x^2-10x+25$ The square of the difference of two monomials is equal to: - the square of the first, - minus twice the product of the first by the second, - plus the square of the second. $(a-b)^2= a^2 - 2ab + b^2$ Example: - $(3x-4)^2 = (3x)^2 - 2 * 3x * 4 + 4^2 = 9x^2 - 24x + 16$ ### Sum times diference - $(2x+5)(2x-5)=(2x)^2-5^2 = 4x^2 - 25$ The sum of two monomials by their difference is equal to the difference of their squares. $(a+b)(a-b)= a²-b²$ Example: - $(3x+4). (3x-4) = (3x)^2 -4^2 =9x^2-16$