UNIT5. ALGEBRA Contenido 1. Algebraic expressions.... 1 Worksheet: algebraic expressions.... 2 2. Monomials.... 3 Worksheet: monomials.... 5 3. Polynomials... 6 Worksheet: polynomials... 9 4. Factorising.... 11 Worksheet: factorising... 12 5. Three algebraic identities... 13 Worksheet: three algebraic identities... 15 1. Algebraic expressions. Vocabulary Branch: rama To deal: negociar, tartar To develop: desarrollar To carry out: llevar a cabo, cumplir To depend on: depender de Algebra is a branch of mathematics that deals with the study of the expressions with letters and numbers. These letters are used to represent quantities so that we can use letters for the arithmetical operations such as +,,, and the power. What do you do when you want to refer to a number that you do not know? Suppose you wanted to refer to the number of buildings in your town, but 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. For example: The triple of a unknown number: 3n The triple of a unknown number minus five units: 3n 5 The following unknown number to x: x + 1 The preceding number to y: y 1 Bilingual Section Page 1
An even number: 2a An odd number: 2z + 1 or 2z 1 An algebraic expression consists of terms (each set of letters or numbers and letters) that are written with arithmetical signs between them which include the addition, subtraction, division and multiplication signs. For example: 3x² + 2x 1. The value of an algebraic expression or evaluate a algebraic expression is the number that we obtain when we substitute numbers for letters in the algebraic expression and carry out all the operations occurring in the expression. So the value of an algebraic expression depends on what number/s we put in place of a letter/s. For example: evaluate 2yx 3 x 2 4ya + 2 at x = 3 y=1 a=0 Plug in 3 for x, 1 for y, and, 0 for a. Remembering to be careful with brackets and negatives: 2(1).( 3) 3 ( 3) 2 4(1).(0) + 2= 2(1).( 27) (9) + 0 + 2= 54 9 + 2= 63 + 2= 61 Always remember to be careful with the minus signs! Worksheet: algebraic expressions. 1. Find the expression: a) Start with x, double it and then subtract 6. b) Start with x, add 4 and then square the result. c) Start with x, take away 5, double the result and then divide by 3. d) Start with x, multiply by 4 and then subtract t. e) Start with x, add y and then double the result. f) Start with a, double it and then add b. g) Start with n, square it and then subtract n. h) Start with x, add 2 and then square the result. i) Brick weighs x kg. How much do 6 bricks weigh? How much do n bricks weigh? j) A man shares x between n children. How much does each child receive? k) Twice of a number l) Quarter of a number m) Two times a number minus this number n) The difference between 832 and a number 2. Evaluate the algebraic expression: yx 3 2zx 2 + 3xz 4 5zx 3 + 2xy 3x 5 y 2x 2 3xz 1 Bilingual Section Page 2
3z + 2x + 3 7 yx 3 at the given values of x, y, z: a) x = 0 y=1 z=2 b) x = -1 y=0 z=-2 c) x = 1 y=-1 z=1 2. Monomials. A monomial is an algebraic expression consisting of only one term. It has two different parts, a known value, the coefficient, multiplied by one or some unknown values represented by letters with exponents that must be constant and positive whole numbers, the literal part. Not a monomial 1+2x Reason It s an algebraic expression but it is not a monomial. A sum is not a monomial 1 A monomial cannot have a variable in the x denominator 5 x A term cannot have a variable in the exponent a -1 3 xy The variable cannot have a negative exponent It is not a monomial because the exponent is not an integer. If the literal part of a monomial has only one letter, then the degree of the monomial is the exponent of the letter. If the literal part of a monomial has more than one letter, then the degree is the sum of the exponents of every letter. For example: The degree of 2x 3 is 3 The degree of 3x 5 y 2 is 5 + 2 = 7. The degree of 7x 3 yz is 3 + 1 + 1 = 5. Bilingual Section Page 3
ADDITION AND SUBTRACTION OF MONOMIALS You can add monomials only if they have the same literal part (they are also called like terms). In this case, you sum the coefficients and leave the same literal part. Like terms use exactly the same literal part. For example: 3x and 6x are like terms. 3x and 3xy are unlike terms. 3x and 3x 2 are unlike terms. Look at these examples: 4xy 2 +3xy 2 = 7xy 2 5x 2 +3 2x 2 +1 = 3x 2 +4, and we can not add the terms 3x 2 and 4 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 variables: For example: a) (2xy 2 ).( -5x 2 y) = -10x 3 y 3 b) 3x 2.2xy = 6x 3 y QUOTIENT OF MONOMIALS. If you want to divide two monomials, you just have to divide the coefficients, and subtract the exponents of the equal variables. The quotient of monomials gives an expression that is not always a monomial. You can also simplify the fractions that result from the division. Look at these examples: (10x 3 ) : (2x) = 5x 2 8 6......... This expression is not a monomial. Bilingual Section Page 4
Worksheet: monomials. 3. Complete the table 4. Name the variables, coefficient, literal part and degree of the following monomials a) 5xy b) 2x c) 5x 2 y 3 d) xy 2 e) x 2 yz f) 2 3 x x g) 5 x 7x 5 y 8 h) x 3 3 4 2 4 i) x y z t j) 8 k) 0x 2 y 7 5. Collect like terms to simplify each expression: 7. Operate Bilingual Section Page 5
3. Polynomials A polynomial is the addition or subtraction of two or more monomials. If there are two monomials, it is called a binomial, for example x² + x If there are three monomials, it is called a trinomial, for example 2x² -3x + 1 The following are NOT polynomials: 4 3 2 The degree of the entire polynomial is the degree of the highest-degree term that it contains, so: x 2 + 2x 7 is a second-degree trinomial x 4 7x 3 is a fourth-degree binomial. The polynomial that follows is a second-degree polynomial, and there are three terms: 4x² is the leading term, and (-7) is the constant term Leading term Constant term 4 3 7 Polynomials are usually written this way, with the terms written in "decreasing" order; that is, with the highest exponent first, the next highest next, and so forth, until you get down to the constant term. Polynomials are also sometimes named for their degree: a second-degree polynomial, such as 4x 2, x 2 9, or ax 2 + bx + c, is also called a "quadratic" a third-degree polynomial, such as 6x 3 or x 3 27, is also called a "cubic" a fourth-degree polynomial, such as x 4 or 2x 4 3x2 + 9, is sometimes called a "quartic" Bilingual Section Page 6
The value of a polynomial or evaluate a polynomial is the number that we obtain when we substitute numbers for letters in the polynomial and carry out all the operations occurring in the expression. So the value of a polynomial depends on what number/s we put in place of a letter/s. For example: Evaluate 2x 3 x 2 4x + 2 at x = 3 Plug in 3 for x, remembering to be careful with brackets and negatives. 2( 3) 3 ( 3) 2 4( 3) + 2= 2( 27) (9) + 12 + 2= 54 9 + 14= 63 + 14= 49 Always remember to be careful with the minus signs! ADDING POLYNOMIALS When adding polynomials you must add each like term of the polynomials, that is, monomials that have the same literal part, you use what you know about the addition of monomials. There are two ways of doing it. The format you use, horizontal or vertical, is a matter of preference (unless the instructions explicitly tell you otherwise). Given a choice, you should use whichever format you're more comfortable with. Note that, for simple additions, horizontal addition (so you don't have to rewrite the problem) is probably the simplest, but, once the polynomials get complicated, vertical addition is probably the safest (so you don't "drop", or lose, terms and minus signs). Here is an example: Horizontally Vertically Either way, I get the same answer. SUBTRACTING POLYNOMIALS When subtracting polynomials you must realize that a subtraction is the addition of the first term and the opposite of the second: Bilingual Section Page 7
A B = A + ( - B) Notice that running the negative through the brackets changes the sign on each term inside the brackets. Look at this example: Horizontally Vertically: write out the polynomials, leaving gaps when necessary, and change the signs in the second line, then add: Either way, I get the same answer. MULTIPLYING POLYNOMIALS The very simplest case for polynomial multiplication is the product of two one-term polynomials. I've already done this type of multiplication when I was first learning to multiply monomials. (5x 2 )( 2x 3 ) = 10x 5 The next step up in complexity is a one-term polynomial times a multi-term polynomial. To do this, I have to distribute the monomial through the brackets. For example: 3 2 3 3.2 3 6 3 5 3. 3.5 3 15 3 4 10 3 4 3 3 10 12 3 30 The next step up is a multi - term polynomial times a multi - term polynomial. The first way I can do this is "horizontally"; in this case, however, I'll have to distribute twice, taking each of the terms in the first parentheses "through" each of the terms in the second parentheses: 3 2. 3..2 3.2 3 2 6 5 6 Bilingual Section Page 8
The "vertical" method is much simpler, because it is similar to the multiplications learnt at primary school, here you can find two examples: 4 3 4 4 7 3 4 4 4 7 3 12 12 12 21 7 12 4 8 19 21 Worksheet: polynomials 8. Name the variables, degree, principal term, constant term of the following polynomials 9. Evaluate the algebraic expressions: x 3 2x 2 + 3x 4, 5x 3 + 2x, 3x 5 2x 2 3x 1 and 3 + 2x + 3 7 x 3 at the given values of x: a) x = 0 b) x = 1 c) x = 2 d) x = -1 10. Calculate the following additions of polynomials: Bilingual Section Page 9
11. Calculate the following subtractions: 12. Expand the brackets: Bilingual Section Page 10
13. Multiply these brackets: 4. Factorising. Factoring is to write an expression as a product of factors. For example, we can write 10 as 5.2, where 5 and 2 are called factors of 10. We can also do this with monomial expressions. 14x 2 = 2.7. x. x Factorize is the reverse process of expanding brackets. The factorized expression has a polynomial inside a bracket, and a term outside. This term outside must be a common term, and it can be either a number or a letter. It means that the number or the letter (s) can be found in every term of the polynomial. Basically, when we factor, we reverse the process of multiplying the polynomial, so that, we are looking for simpler polynomials that can be multiplied together to give us the polynomial that we started with. The trick is to see what can be factored out of every term in the expression. Just don't make the mistake of thinking that "factoring" means "dividing off and making disappear". Nothing disappears when you factor; things just get rearranged. For example: factor 2y+6 Firstly we factor "splitting" (separar) an expression into a multiplication of simpler expressions each term: 2y = 2 y and 6 = 2 3 Both 2y and 6 have a common factor of 2. Bilingual Section Page 11
Secondly we rearrange the common term in front of a bracket 2y + 6 = 2. (y+3) So, 2y+6 has been "factored into" 2 and y+3 Finally check it: 2(y+3) = 2y+6 Note that factoring is the opposite of expanding: expanding 2 3 2 6 factoring In the previous example we saw that 2y and 6 have a common factor: 2 But, to make sure that you have done the job properly you need to make sure you have the highest common factor, including any variables. For example: factor 3y 2 +12y Firstly, 3 and 12 have a common factor of 3. But 3y 2 and 12y also share the variable y. Together that makes 3y: y 2 = 3y. y and 12y = 3y. 4 So you can factor the whole expression into 3y 2 +12y = 3y.(y+4) Check it: 3y(y+4) = 3yy + 3y. 4 = 3y 2 +12y Here are some examples of how to factor: 3x 12 = 3.(x 4) 12y 2 5y = y.(12y 5) 3x 3 + 6x 2 15x = 3x.(x 2 + 2x 5) Remember: when the term to be factored out coincides with one of the addends, the unit always remains: Worksheet: factorising 14. Factorize:. 1 Bilingual Section Page 12
15. Factorize: 5. Three algebraic identities There are three formulas about operations with binomials that are very common and it is useful to memorize. Square of an addition. We want to find a formula to work out (a + b) 2 where a and b can be numbers or any kind of monomial. 2 Note that (a + b) 2 is not the same as a 2 + b 2 so don t forget the term 2ab Square of a subtraction. We want to find a formula to work out (a - b) 2 where a and b can be numbers or any kind of monomial. 2 Bilingual Section Page 13
Note again that (a b) 2 is not the same as a 2 b 2 but something absolutely different. Difference of squares What happens if we multiply (a + b)(a b)? The terms ab and ab cancel out, so (a + b)(a b) = a 2 b 2 We can use these three formulas in two directions to expand or to factorise any algebraic expression that fits with any of them. Remembering these identities can make factoring easier. For example: factor 4x 2-9 I can't see any common factors. But if you know your Special Binomial Products you could recognize it as the difference of squares, because 4x 2 is (2x) 2, and 9 is (3) 2, so we have: 4 9 2 3 And that can be produced by the difference of squares formula: Where a is 2x, and b is 3.. Let us try doing that: 2 3. 2 3 2 3 4 9 So the factors of 4 9 are 2 3 and 2 3. When trying to factor, follow these steps: "Factor out" any common terms See if it fits any of the identities, plus any more you may know Keep going till you can't factor any more Bilingual Section Page 14
Worksheet: three algebraic identities 16. Calculate using a formula. 17. Factorise: Bilingual Section Page 15