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Index Monomials Operations Polinomials Operations with polynomials Multiplying polynomials Exercises Algebraic expressions Matem´aticas 2o E.S.O. Alberto Pardo…
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Index Monomials Operations Polinomials Operations with polynomials Multiplying polynomials Exercises Algebraic expressions Matem´aticas 2o E.S.O. Alberto Pardo Milan´es Index Monomials Operations Polinomials Operations with polynomials Multiplying polynomials Exercises 1 Monomials 2 Operations with monomials 3 Polinomials 4 Operations with polynomials 5 Multiplying polynomials 6 Exercises Alberto Pardo Milan´es Algebraic expressions Index Monomials Operations Polinomials Operations with polynomials Multiplying polynomials Exercises Monomials Alberto Pardo Milan´es Algebraic expressions Index Monomials Operations Polinomials Operations with polynomials Multiplying polynomials Exercises Monomials Numerical value of an algebraic expression An algebraic expression in variables x, y, z, a, r, t . . . k is an expression constructed with the variables and numbers using addition, multiplication, and powers. To evaluate the numerical value of an algebraic expression means that you have to replace the variable in the expression with values and simplify the expression. Example: To find the value of the algebraic expression x2 −3x+4 if x = 3, you replace every x by 3 and simplify: 32 − 3 · 3 + 4 = 9 − 9 + 4 = 4. Alberto Pardo Milan´es Algebraic expressions Index Monomials Operations Polinomials Operations with polynomials Multiplying polynomials Exercises Monomials A number multiplied with a variable in an algebraic expression is named coefficient. The product of a coefficient and one or more variables is called a monomial. Examples: x, 3xy2 and 2 5 x2 y3 z are all monomials, the coefficients are 1, 3 and 2 5 . In a monomial with only one variable, the power is called its order, or sometimes its degree. In a monomial with several variables, the order/degree is the sum of the powers. Examples: Deg(2x4)=4, Deg(7x3y2)=5. Like monomials are monomials that have the exact same variables, but different coefficients. Unlike monomials are monomials that are not like monomials. Examples: 2x3y2 and 2 5 x3 y2 are like monomials. 4xy2 and 4y2x4 are unlike monomials. Alberto Pardo Milan´es Algebraic expressions Index Monomials Operations Polinomials Operations with polynomials Multiplying polynomials Exercises Operations with monomials Alberto Pardo Milan´es Algebraic expressions Index Monomials Operations Polinomials Operations with polynomials Multiplying polynomials Exercises Operations with monomials (Adding and subtracting monomials) You can ONLY add and subtract like monomials. To add or subtract like monomials, add or subtract the coefficients and keep the variables. Examples: 3x+4x = (3+4)x = 7x, and 20a−24a = (20−24)a = = −4a. (Multiplying monomials) To multiply monomials, multiply the coefficients and add the exponents with the same bases. Examples: 3x2 · 5y = (3 · 5)x2 · y = 15x2y, and 2a2 · 7ab4 = = (2 · 7)a2 · ab4 = 14a2+1b4 = 14a3b4. Alberto Pardo Milan´es Algebraic expressions Index Monomials Operations Polinomials Operations with polynomials Multiplying polynomials Exercises Operations with monomials (Dividing monomials) To divide monomials, divide the coefficients and subtract the exponents with the same bases. Example: 15x3y3z2 : 3xy3z = (15 : 3)x3−1y3−3z2−1 = 5x2z. Alberto Pardo Milan´es Algebraic expressions Index Monomials Operations Polinomials Operations with polynomials Multiplying polynomials Exercises Polinomials Alberto Pardo Milan´es Algebraic expressions Index Monomials Operations Polinomials Operations with polynomials Multiplying polynomials Exercises Polinomials A polynomial is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. A polynomial in one variable is given by a sum of several monomials. Example: 3x2 − 5x − 2. In a polynomial with only one variable, the highest power is called its order, or sometimes its degree. Example: Deg(x2 + 3x4 − 2x3 + 1) = Deg(3x4) = 4. In a polynomial with several variables, the order/degree is the highest sum of the powers of every term. Example: Deg(xz3 + 3yx2z2 − 2) = Deg(yx2z2) = 5. Alberto Pardo Milan´es Algebraic expressions Index Monomials Operations Polinomials Operations with polynomials Multiplying polynomials Exercises Operations with polynomials Alberto Pardo Milan´es Algebraic expressions Index Monomials Operations Polinomials Operations with polynomials Multiplying polynomials Exercises Operations with polynomials Add and subtract (Adding polynomials) Adding polynomials is just a matter of combining like monomials. Remenber you can only add like monomials. Example: (7x2−x+4)+(x2−2x+3) = 7x2−x+4+x2−2x+3 = = 9x2 − 3x + 7. (Subtracting polynomials) To subtract a polynomial use the opposite of every coefficient of the subtrahend and add like monomials. Example: (x3 + 3x2 + 5x − 4) − (3x3 − 8x2 − 5x + 6) = = x3 + 3x2 + 5x − 4 − 3x3 + 8x2 + 5x − 6 = = −2x3 + 11x2 + 10x − 10 Alberto Pardo Milan´es Algebraic expressions Index Monomials Operations Polinomials Operations with polynomials Multiplying polynomials Exercises Multiplying polynomials Alberto Pardo Milan´es Algebraic expressions Index Monomials Operations Polinomials Operations with polynomials Multiplying polynomials Exercises Multiplying polynomials (Multiply polynomials) To multiply two polynomials, we multiply each monomial of one polynomial (with its sign) by each monomial (with its sign) of the other polynomial. Write these products one after the other (with their signs) and then add like monomials to form the complete product. Example: (x + 3)(−2x + 2) = (x + 3)(−2x) + (x + 3)2 = = (−2x2 − 6x) + (2x + 6) = −2x2 − 4x + 6. Alberto Pardo Milan´es Algebraic expressions Index Monomials Operations Polinomials Operations with polynomials Multiplying polynomials Exercises Multiplying polynomials Multiply it vertically Sometimes doing it vertically can be nicer: Example: (4x2 − 4x − 7)(x + 3) = (4x2 − 4x − 7)x + (4x2 − 4x − 7)3 = 4x3 − 4x2 − 7x + 12x2 − 12x − 21 = 4x3 + 8x2 − 19x − 21 4x2 − 4x − 7 × x + 3 12x2 − 12x − 21 4x3 − 4x2 − 7x 4x3 + 8x2 − 19x − 21 Alberto Pardo Milan´es Algebraic expressions Index Monomials Operations Polinomials Operations with polynomials Multiplying polynomials Exercises Exercises Alberto Pardo Milan´es Algebraic expressions Index Monomials Operations Polinomials Operations with polynomials Multiplying polynomials Exercises Exercises Exercise 1 Convert the statements into an algebraic expression using a variable and a sum or a difference: • A number plus four: • Five more than a number: • A number minus five: • The sum of a number and two: • A number increased by ten: • One less than a number: • Seven added to a number: • The difference of a number and eight: • Nine less than a number • A number decreased by three: • Six subtracted from a number: • The age a boy was two years ago: Alberto Pardo Milan´es Algebraic expressions Index Monomials Operations Polinomials Operations with polynomials Multiplying polynomials Exercises Exercises Exercise 2 Convert the statements into an algebraic expression using a variable and a multiplication or a division: • Double a number: • The quotient of a number and six: • The product of four and a number: • Twice a number: • Nine divided by a number: • A number multiplied by negative five: • One fifth of a number: • Three times a number: • The ratio of a number to four: • Eighty percent of a number : Alberto Pardo Milan´es Algebraic expressions Index Monomials Operations Polinomials Operations with polynomials Multiplying polynomials Exercises Exercises Exercise 3 Write the sentence as an algebraic expression: • My bedroom’s lenght is 2 more feet than its width n. The lenght is . . . • The temperature at noon was t and had risen 8 degrees since seven o’clock. The temperature at 7:00 was . . . • Lou charges 6,50 euros an hour to baby-sit. Today he worked x hours which means that he earned . . . • I have y stamps from Asia and I have seven fewer stamps from Europe than from Asia. The total number of stamps I have is . . . Alberto Pardo Milan´es Algebraic expressions Index Monomials Operations Polinomials Operations with polynomials Multiplying polynomials Exercises Exercises Exercise 4 Write the sentence as an algebraic expression and operate: • The base of a rectangle is double than the height. The area of the rectangle is. . . • The product of a number and the number than comes after it is. . . • I have nine fewer coins from China than from Australia. The total number of coins I have is . . . • Tom’s age is double than Fred’s age. The product of their ages is. . . • The sum of a number and twice the number that comes before it is. . . Alberto Pardo Milan´es Algebraic expressions Index Monomials Operations Polinomials Operations with polynomials Multiplying polynomials Exercises Exercises Exercise 5 Find the degree of these monomials: Deg(5x4) = Deg(4y) = Deg( 1 2 z3 ) = Deg(abch2) = Deg(4xy) = Deg(x2) = Deg(33x7) = Deg(5a4b) = Deg( 3 5 x2 y) = Deg(3x2y2) = Alberto Pardo Milan´es Algebraic expressions Index Monomials Operations Polinomials Operations with polynomials Multiplying polynomials Exercises Exercises Exercise 6 Link like monomials: 7x 4x2y 2xy 7xy 3 5 x2 y 1 2 x 3xy2 2xy2 Alberto Pardo Milan´es Algebraic expressions Index Monomials Operations Polinomials Operations with polynomials Multiplying polynomials Exercises Exercises Exercise 7 Complete: Monomial Coefficient Order Variables −7ab2 3 2 m3 n2 p2 −7 √ 3x3y4 Alberto Pardo Milan´es Algebraic expressions Index Monomials Operations Polinomials Operations with polynomials Multiplying polynomials Exercises Exercises Exercise 8 Find: 2x2 · 3y = 2x2y3 · 7 4 x2 y = 4xa : 10a = 3 5 x · 15y = 5xa · 10yb = 9 2 x2 y3 : 3 2 xy = 1 2 x2 y · 3y = 4z · 5zy = 9zab2 : 6z = Alberto Pardo Milan´es Algebraic expressions Index Monomials Operations Polinomials Operations with polynomials Multiplying polynomials Exercises Exercises Exercise 9 Find the degree of these polynomials: Deg(x4 − 3x3 + 2x2 + 1) = Deg(x2 − 3x3 + 2x + 1) = Deg(xy − x2 + yx3 + y2) = Deg(x4y2 − 3x5) = Alberto Pardo Milan´es Algebraic expressions Index Monomials Operations Polinomials Operations with polynomials Multiplying polynomials Exercises Exercises Exercise 10 Find: (4x3 − 2x + 5) + (x2 − 2x + 1) + (−4x2 − 6) = (7x3 + 2x2 + 4x + 9) − (2x3 − 3x + 8) = (2x − 3) · (x + 12) = (x3 + 3x2 + 3x + 1) · (x2 + 1) = Alberto Pardo Milan´es Algebraic expressions

Beamer14

Sep 8, 2017
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