Algebra
- 1. I.E.S. MARÍA BELLIDO - BAILÉN BILINGUAL SECTION – MARÍA ESTHER DE LA ROSA ALGEBRA 1. ALGEBRAIC EXPRESSIONS Algebra is the handling of numerical relations in which one or more quantities are unknown. These terms are called variables or unknowns and they are represented by letters. An algebraic expression is a combination of letters and numbers linked by the signs of operations: addition, subtraction, multiplication and powers. EXAMPLES: The algebraic expressions allow us, for example to find areas and volumes. Length of the circumference: L = 2 r, where r is the radius of the circumference. Area of the square: S = l2, where l is the side of the square. EXAMPLES OF COMMON ALGEBRAIC EXPRESSIONS: The double of a number: 2x. The triple a number: 3x Half of a number: x/2. A third of a number: x/3. A quarter of a number: x/4. A number to the square: x2 Two consecutive numbers: x and x + 1. 2. NUMERICAL VALUE OF AN ALGEBRAIC EXPRESSION The numerical value of an algebraic expression for a particular value is the number obtained by replacing the unknown values with the numerical value given and perform the operations. EXAMPLE: f(x)= x3 x = 5 cm f(5) = 53 = 125 1
- 2. 3. TYPES OF ALGEBRAIC EXPRESSIONS A monomial is an algebraic expression formed by a single term, in which the only operations that appear between the variables are the product and the power of a natural exponent. Example: 2x2y3z. A binomial is an algebraic expression formed by two terms. A polynomial is an algebraic expression consists of more than one term. 4. PARTS OF A MONOMIAL The coefficient of a monomial is the number that multiplies the variable(s). The literal part is constituted by the letters and its exponents. The degree of a monomial is the sum of all exponents of the letters or variables. gr(2x2 y3 z) = 2 + 3 + 1 = 6 Similar monomials: Two monomials are similar when they have the same literal part. 2x2 y3 z is similar to 5x2 y3 z 5. OPERATIONS WITH MONOMIALS 5.1 Addition of Monomials: We can add monomials if they are similar. The sum of the monomials is another monomial that has the same literal part and whose coefficient is the sum of the coefficients. EXAMPLE: 2x2 y3 z + 3x2 y3 z = 5x2 y3 z If the monomials are not similar a polynomial is obtained: x2 y3 + 3x2 y3 z 5.2Multiplication of a Number by a Monomial The product of a number by a monomial is another similar monomial whose coefficient is the product of the coefficient of the monomial and the number. EXAMPLE: 5 · (2x2 y3 z) = 10x2 y3 z 2
- 3. 5.3 Multiplication of Monomials The multiplication of monomials is another monomial that takes as its coefficient the product of the coefficients and whose literal part is obtained by multiplying the powers that have the same base. EXAMPLE: (5x2 y3 z) · (2 y2 z2) = 10 x2 y5 z3 5.4 Division of Monomials Dividing monomials can only be performed if they have the same literal part and the degree of the dividend has to be greater than or equal to the corresponding divisor. The division of monomials is another monomial whose coefficient is the quotient of the coefficients and its literal part is obtained by dividing the powers that have the same base. EXAMPLE: If the degree of divisor is greater, an algebraic fraction is obtained. EXAMPLE: 5.5 Power of a Monomial To determine the power of a monomial, every element in the monomial is raised to the exponent of the power. EXAMPLE: (2x3)3 = 23 · (x3)3 = 8x9 (−3x2)3 = (−3)3 · (x2)3 = −27x6 6. POLYNOMIALS A polynomial is an algebraic expression in the form: P(x) = an xn + an - 1 xn - 1 + an - 2 xn - 2 + ... + a1 x1 + a0 an, an -1 ... a1 , ao... are the numbers and are called coefficients n is a natural number. x is the variable. ao is the independent term. The degree of a polynomial P(x) is the greatest degree of the monomials. Two polynomials are similar if they have the same literal part. EXAMPLE: P(x) = 2x3 + 5x − 3 and Q(x) = 5x3 − 2x − 7 3
- 4. 7.OPERATIONS WITH POLYNOMIALS 6.1 Adding Polynomials To add two polynomials, add the coefficients of the terms of the same degree. EXAMPLE: P(x) = 2x3 + 5x − 3 Q(x) = 2x3 − 3x2 + 4x P(x) + Q(x) = (2x3 + 5x − 3) + (2x3 − 3x2 + 4x) = 2x3 + 2x3 − 3x2 + 5x + 4x − 3= 4x3 − 3x2 + 9x − 3 Polynomials can also be added by writing them under each other, so that similar monomials that are in the same columns can be added together. P(x) = 7x4 + 4x2 + 7x + 2 Q(x) = 6x3 + 8x +3 P(x) + Q(x) = 7x4 + 6x3 + 4x2 + 15x + 5 6.2 Multiplication of a Number by a Polynomial It is another polynomial that has the same degree. The coefficients are the product of the coefficients of the polynomial and the number. EXAMPLE: 3 · (2x3 − 3x2 + 4x − 2) = 6x3 − 9x2 + 12x − 6 6.3Multiplying a monomial by a Polynomial The monomial is multiplied by each and every one of the monomials that form the polynomial. EXAMPLE: 3x2 · (2x3 − 3x2 + 4x − 2) = 6x5 − 9x4 + 12x3 − 6x2 6.4Multiplication of Polynomials Multiply each monomial from the first polynomial by each of the monomials in the second polynomial. The multiplication of polynomials is another polynomial whose degree is the sum of the degrees of the polynomials that are to be multiplied. EXAMPLE: P(x) = 2x2 − 3 Q(x) = 2x3 − 3x2 + 4x P(x) · Q(x) = (2x2 − 3) · (2x3 − 3x2 + 4x) = = 4x5 − 6x4 + 8x3 − 6x3 + 9x2 − 12x = 4x5 − 6x4 + 2x3 + 9x2 − 12x The polynomials can also be multiplied as follows: 4
- 5. 8. ALGEBRAIC IDENTITIES Square of a Binomial: (a ± b)2 = a2 ± 2 · a · b + b2 EXAMPLES : (x + 3)2 = x 2 + 2 · x ·3 + 32 = x 2 + 6 x + 9 (2x − 3)2 = (2x)2 − 2 · 2x · 3 + 32 = 4x2 − 12 x + 9 Difference of Squares: (a + b) · (a − b) = a2 − b2 EXAMPLE: (2x + 5) · (2x - 5) = (2x)2 − 52 = 4x2 − 25 Cube of a Binomial: (a ± b)3 = a3 ± 3 · a2 · b + 3 · a · b2 ± b3 EXAMPLES: (x + 3)3 = x3 + 3 · x2 · 3 + 3 · x · 32 + 33 = x 3 + 9x2 + 27x + 27 (2x − 3)3 = (2x)3 − 3 · (2x)2 ·3 + 3 · 2x · 32 − 33 = 8x 3 − 36x2 + 54x − 27 Square of a Trinomial: (a + b + c)2 = a2 + b2 + c2 + 2 · a · b + + 2 · a · c + 2 · b · c EXAMPLE: (x2 − x + 1)2 = (x2)2 + (−x)2 + 12 + 2 · x2 · (−x) + 2 x2 · 1 + 2 · (−x) · 1= x4 + x2 + 1 − 2x3 + 2x2 − 2x= x4− 2x3 + 3x2 − 2x + 1 9. METHODS FOR FACTORING A POLYNOMIAL 9.1.Remove the Common Factor: It consists by applying the distributive property. a · b + a · c + a · d = a (b + c + d) EXAMPLES. x3 + x2 = x2 (x + 1) 2x4 + 4x2 = 2x2 (x2 + 2) x2 − ax − bx + ab = x (x − a) − b (x − a) = (x − a) · (x − b) 5
- 6. 9.2Remarkable Identities: A difference of squares is equal to the sum of the difference. a2 − b2 = (a + b) · (a − b) EXAMPLES: x2 − 4 = (x + 2) · (x − 2) x4 − 16 = (x2 + 4) · (x2 − 4) = (x + 2) · (x − 2) · (x2 + 4) A perfect square trinomial is equal to a squared binomial. a2 ± 2 a b + b2 = (a ± b)2 . 6