UNIT 5 VOCABULARY: POLYNOMIALS


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1 2º ESO Bilingüe Page 1 UNIT 5 VOCABULARY: POLYNOMIALS 1.1. Algebraic Language Algebra is a part of mathematics in which symbols, usually letters of the alphabet, represent numbers. Letters are used to represent quantities, so that we can express relationships between them by using the arithmetical, operations such as +,,, and exponents. What do you do when you want to refer to a number that you do not know? Suppose you want to refer to the number of buildings in your town, but you haven't counted them yet. You could say 'blank' number of buildings, or perhaps '?' number of buildings. In mathematics, letters are often used to represent numbers that we do not know  so you could say 'x' number of buildings, or 'q' number of buildings. These are called variables. Look at these examples: The triple of a number: 3n The triple of a number minus five units: 3n 5 The following number to x: x + 1 The preceding number to y: y 1 An even number: 2a An odd number: 2z + 1 or 2z Algebraic expressions An algebraic expression is a combination of numbers, variables, brackets, connected with operations. Exercises. Find the algebraic expression for this sentences: Sentence Algebraic Expression I start with x, double it and then subtract 6. I start with x, add 4 and then square the result. I start with x, take away 5, double the result and then divide by 3. I start with x, multiply by 4 and then subtract t. I start with x, add y and then double the result. I start with a, double it and then add b.
2 2º ESO Bilingüe Page 2 I start with n, square it and then subtract n. I start with x, add 2 and then square the result. A brick weighs x kg. How much do 6 bricks weigh? How much do n bricks weigh? A man shares x euros between n children. How much does each child receive? 1.3. Monomials A monomial is an algebraic expression consisting of only one term. A monomial can be any of the following: A constant: A variable: x y z 2 3 are monomials. The product of a constant and one or more variables: 2a 3xy 5x³ x²y² THE VARIABLES OF A MONOMIAL SHOULD NOT HAVE NEGATIVE OR FRACTIONAL EXPONENTS! Therefore, these algebraic expressions are not monomials: 3x 2 3xy 1/2 3 t 2 The coefficient of a monomial is the number that multiplies the variable(s). The coefficient of 3x is 3. The degree of x 2 y 2 z is 1 (because =5). The literal part of a monomial is formed by the variables (letters) and its exponents. The literal part of 3x is x. The literal part of x 2 y 2 z is x²y²z. The degree of a monomial is defined as the sum of all the exponents of the variables. The degree of 3x is 1 (the exponent of x is 1). The degree of x 2 y 2 z is 5 (because =5). The degree of 6 is 0 (there is no variable). Exercises. 1. Name the variables, coefficient, literal part and degree of the following monomials: 1. 5xy b) 2x c) xy² d) x²yz e) 3 7 x 5 y 8 f) 8
3 2º ESO Bilingüe Page 3 2. Complete the table: Monomial Variables Coefficient Literal part Degree 3x 2 y 3 7x 3 yz 4 5 x3 3 2 x Polynomials A polynomial is an expression which is made up of monomials that are added or subtracted. Polynomials can have: No variable at all (for example, 21 is a polynomial that has just one term, which is a constant). One variable (for example x 4 2x 2 + x has three terms, but only one variable, x). Two or more variables (for example, xy 45x 2 z has two terms, and three variables,x, y and z). The degree of a polynomial with only one variable is the largest exponent of that variable. A term that doesn't have a variable in it is called constant term. The coefficient of the term with the highest degree is called the leading coefficient. Exercises. 1. Complete this table: Polynomial Degree Leading coefficient Constant term x 4 3x 5 + 2x 2 + 5
4 2º ESO Bilingüe Page 4 1 x 4 x 2 2 x x 3 x Evaluating a polynomial To evaluate a polynomial, you plug in (substitute) the given value of x, and calculate the value. For example, to evaluate P(x) = 2x 3 x 2 4x + 2 at x = 3: P( 3) = 2 ( 3) 3 ( 3) 2 4 ( 3) + 2 = 2 ( 27) (9) = = = 49 ALWAYS REMEMBER TO BE CAREFUL WITH BRACKETS AND THE MINUS SIGNS! Exercise. Evaluate x 4 + 3x 3 x a) for x = 3 b) for x = 3 c) for x = 0 d) for x= Addition and subtraction of monomials You can add or subtract monomials only if they have the same literal part (they are also called like terms). In this case, you sum or subtract the coefficients and leave the same literal part. You can find some examples of like terms in the picture: Look at these examples: 4xy² + 3xy² = 7xy² (we can add these monomials because they have the same literal part). 3xz + 7xz (we can't add these monomials because they don't have the same literal part). 2x² + 3 5x² + 1 = 3x² + 4 (and we can't add 3x² and 4 because they don't have the same literal part). Exercise. Collect like terms to simplify each expression: a) x² + 3x x + 3x² c) (2x + 3) (5x 7) (x 1) e) (x² + 2x) (2x² x) + (3x² + 5x) 2 b) 5y + 3x + 2y + 4x d) 3 x x 3 2 x2 1 5 x+2 f) (x² + y) + (7x² 3y) (x² + 7y)
5 2º ESO Bilingüe Page Product of monomials If you want to multiply two or more monomials, you just have to multiply the coefficients, and add the exponents of the equal letters: More examples: 2xy² (5x²y) = 10 x³ y³ 3a² 2ab = 6a³ b Exercise. Multiply: 2 a) 5x 3x c) 2x x³ e) 7x²y xy² g) 3 x3 y 3x i) x² (2x) 3x³ b) 2x 3x² d) 4xy 2x²y f) 5x³y² xy h) 3x 2x² 5x³ j) 3ab (2a²) 2.3. Quotient of monomials If you want to divide two monomials, you just have to divide the coefficients, and subtract the exponents of the equal letters. THE QUOTIENT OF MONOMIALS GIVES AN EXPRESSION THAT IS NOT ALWAYS A MONOMIAL! More examples: 10x³ : 2x = 5x² 8x 2 y : 6 y 3 = 8x2 y 6 y = 4x 2 3 3y 2 Exercise. Operate: a) 15x³ : 3x² c) 6x : 2x³ e) 15a³b² : ab g) 5xy 2x²y : 3x b) 2x 4 : 3x d) 12a²b : 3a f) 3x 5 y 2 :3x 2 y 2 h) xy³ 3x²y : 2x²y² 3.1. Addition of polynomials To add two polynomials, you have to follow these steps: Place like terms together. Add the like terms For example, to add p(x) = 2x 2 + 6x + 5 and q(x) = 3x 22x  1 Start with: 2x 2 + 6x x 22x  1 = Place like terms together: (2x 2 + 3x 2 ) + (6x  2x) + (51) = Add the like terms: = 5x 2 + 4x + 4
6 2º ESO Bilingüe Page 6 You could also add two polynomials in columns like this: 3.2. Subtraction of polynomials To subtract polynomials, follow these steps: First reverse the sign of each term you are subtracting (in other words, turn "+" into "", and "" into "+"). Add as usual. For example, to subtract p(x) = 2x 2 + 6x + 5 and q(x) = 3x 22x  1 Start with: 2x 2 + 6x (3x 22x  1) = Reverse the signs of each term: 2x 2 + 6x + 53x 2 + 2x + 1 = Add the like terms: = 2x 2 3x 2 + 6x + 2x = x 2 + 8x + 6 And again, you can also do it in columns: Exercise. Given the polynomials: P(x) = 4x 2 1 Q(x) = x 3 3x 2 + 6x 2 R(x) = 6x 2 + x + 1 S(x) = x2 +4 T(x) = 2 x2 +5 U(x) = x Calculate: a) P(x) + Q (x) b) P(x) U(x) c) P(x) + R (x) d) 2P(x) R(x) e) S(x) + T(x) + U(x) f) S(x) T(x) + U(x) 3.3. Multiplication of polynomials Before multiplying polynomials, let us look at a simpler case first: Monomial times binomial Multiply the monomial by each of the two terms, like this: Polynomial times polynomial Follow these steps:
7 2º ESO Bilingüe Page 7 Multiply each term in the first polynomial by each term in the second polynomial. Always remember to add like terms. For example, to multiply (x + 2y) (3x 4y + 5) We multiply each term in the first polynomial by each term in the second polynomial: 3x 2 4xy + 5x + 6xy 8y y We add like terms: 3x 2 + 2xy + 5x 8y y And once again, you can also do it in columns: Exercise. Multiply: a) (x 4 2x 2 + 2) (x 2 2x + 3) b) (3x 2 5x) (2x 3 + 4x 2 x + 2) 4.1. Special products SQUARE OF A SUM! (a+b) 2 =a 2 +2ab+b 2 Examples: (x +5) 2 =x 2 +2 x =x 2 +10x+25 (2x+3) 2 =(2x) x =4x 2 +12x+9 SQUARE OF A DIFFERENCE! (a b) 2 =a 2 2ab+b 2 Examples: (x 2 2) 2 =(x 2 ) 2 2 x =x 4 4x 2 +4 (x 3) 2 =x 2 2 x 3+( 3) 2 =x x +3
8 2º ESO Bilingüe Page 8 SUM TIMES DIFFERENCE! (a+b)(a b)=a 2 b 2 Examples: ( x + 3 2) ( x 3 2) ( =x 2 3 2)2=x (ab 2 +5)(ab 2 5)=(ab 2 ) =a 2 b 4 25 Exercises. 1. Expand the following expressions: a) (x + 2) 2 c) (3x + 7)² e) (x 3)² g) (a + 2b)² b) (x + 5) 2 d) ( x 2 2 ) 2 f) (x² + 2)² h) (2x  2y)² 2. Expand the following expressions: a) (3x 2) (3x + 2) c) (3 + x) (3  x) b) (x + 5) (x 5) d) (x 2) (x+ 2) 3. Express as a binomial: a) x² + 12x + 36 = (x + )² f) 4x² 9 = b) x² 16 = (x + ) (x  ) g) 64x²  160x = c) 4x² 20x + 25 = (  5 )² h) 9x² 25 = d) x² + 14x + 49 = i) 25x² + 40x + 16 = e) x² 1 = j) x 2 x Difference between equation,expression, identity and formulae In Algebra, you use letter symbols to represent unknowns in a variety of situations: EQUATIONS In an equation the letters stand for one or more particular numbers (the solutions of the equation). 2x + 1 = x 2 EXPRESSIONS IDENTITIES In an expression there is no equals sign. 3x² + 2x 1 In an identity there is an equals sign, but the equality holds for all values of the unknown. 2(x + 1) = 2x + 2
9 2º ESO Bilingüe Page 9 FORMULAE In a formula, letters stand for defined quantities or variables. d = s t. (d is distance, s is speed and t is time) Exercise. Separate the equations, the formulae, the identities and the expressions: a) x ( x + 1 ) = x² + x c) V = I R e) 7x + 11 = x 9 g) A = π r² b) 7y + 10 d) x²  3x + 10 f) (x + 1)² = x² + 2x + 1 h) x²  7x = 0
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