120 Examples 8c 8 is the coefficient to the variable c. 6ab 6 is the coefficient to both variables, a and b. THREE KINDS OF POLYNOMIALS ■ Monomials are single terms that are composed of variables and their exponents and a positive or nega- tive coefficient. The following are examples of monomials: x,5x,–6y 3 ,10x 2 y,7,0. ■ Binomials are two non-like monomial terms separated by + or – signs. The following are examples of binomials: x + 2, 3x 2 – 5x,–3xy 2 + 2xy. ■ Trinomials are three non-like monomial terms separated by + or – signs. The following are examples of trinomials: x 2 + 2x – 1, 3x 2 – 5x + 4, –3xy 2 + 2xy – 6x. ■ Monomials, binomials, and trinomials are all examples of polynomials, but we usually reserve the word polynomial for expressions formed by more three terms. ■ The degree of a polynomial is the largest sum of the terms’ exponents. Examples ■ The degree of the trinomial x 2 + 2x – 1 is 2, because the x 2 term has the highest exponent of 2. ■ The degree of the binomial x + 2 is 1, because the x term has the highest exponent of 1. ■ The degree of the binomial –3x 4 y 2 + 2xy is 6, because the x 4 y 2 term has the highest exponent sum of 6. LIKE TERMS If two or more terms have exactly the same variable(s), and these variables are raised to exactly the same expo- nents, they are said to be like terms. Like terms can be simplified when added and subtracted. Examples 7x + 3x = 10x 6y 2 – 4y 2 = 2y 2 3cd 2 + 5c 2 d cannot be simplified. Since the exponent of 2 is on d in 3cd 2 and on c in 5c 2 d, they are not like terms. The process of adding and subtracting like terms is called combining like terms. It is important to combine like terms carefully, making sure that the variables are exactly the same. Algebraic Expressions An algebraic expression is a combination of monomials and operations. The difference between algebraic expres- sions and algebraic equations is that algebraic expressions are evaluated at different given values for variables, while algebraic equations are solved to determine the value of the variable that makes the equation a true statement. There is very little difference between expressions and equations, because equations are nothing more than two expressions set equal to each other. Their usage is subtly different. – THEA MATH REVIEW– Example A mobile phone company charges a $39.99 a month flat fee for the first 600 minutes of calls, with a charge of $.55 for each minute thereafter. Write an algebraic expression for the cost of a month’s mobile phone bill: $39.99 + $.55x,where x represents the number of additional minutes used. Write an equation for the cost (C) of a month’s mobile phone bill: C = $39.99 + $.55x,where x represents the number of additional minutes used. In the above example, you might use the expression $39.99 + $.55x to determine the cost if you are given the value of x by substituting the value for x. You could also use the equation C = $39.99 + $.55x in the same way, but you can also use the equation to determine the value of x if you were given the cost. SIMPLIFYING AND EVALUATING ALGEBRAIC EXPRESSIONS We can use the mobile phone company example above to illustrate how to simplify algebraic expressions. Alge- braic expressions are evaluated by a two-step process; substituting the given value(s) into the expression, and then simplifying the expression by following the order of operations (PEMDAS). Example Using the cost expression $39.99 + $.55x, determine the total cost of a month’s mobile phone bill if the owner made 700 minutes of calls. Let x represent the number of minutes over 600 used, so in order to find out the difference, subtract 700 – 600; x = 100 minutes over 600 used. Substitution: Replace x with its value, using parentheses around the value. $39.99 + $.55x $39.99 + $.55(100) Evaluation: PEMDAS tells us to evaluate Parentheses and Exponents first. There is no operation to perform in the parentheses, and there are no exponents, so the next step is to multiply, and then add. $39.99 + $.55(100) $39.99 + $55 = $94.99 The cost of the mobile phone bill for the month is $94.99. You can evaluate algebraic expressions that contain any number of variables, as long as you are given all of the values for all of the variables. – THEA MATH REVIEW– 121 Simple Rules for Working with Linear Equations A linear equation is an equation whose variables’ highest exponent is 1. It is also called a first-degree equation. An equation is solved by finding the value of an unknown variable. 1. The equal sign separates an equation into two sides. 2. Whenever an operation is performed on one side, the same operation must be performed on the other side. 3. The first goal is to get all of the variable terms on one side and all of the numbers (called constants) on the other side. This is accomplished by undoing the operations that are attaching numbers to the variable, thereby isolating the variable. The operations are always done in reverse “PEMDAS” order: start by adding/subtracting, then multiply/divide. 4. The final step often will be to divide each side by the coefficient, the number in front of the variable, leav- ing the variable alone and equal to a number. Example 5m + 8 = 48 –8 = –8 ᎏ 5 5 m ᎏ = ᎏ 4 5 0 ᎏ m = 8 Undo the addition of 8 by subtracting 8 from both sides of the equation. Then undo the multiplication by 5 by dividing by 5 on both sides of the equation. The variable, m, is now isolated on the left side of the equation, and its value is 8. Checking Solutions to Equations To check an equation, substitute the value of the variable into the original equation. Example To check the solution of the previous equation, substitute the number 8 for the variable m in 5m + 8 = 48. 5(8) + 8 = 48 40 + 8 = 48 48 = 48 Because this statement is true, the answer m = 8 must be correct. – THEA MATH REVIEW– 122 ISOLATING VARIABLES USING FRACTIONS Working with equations that contain fractions is almost exactly the same as working with equations that do not contain variables, except for the final step. The final step when an equation has no fractions is to divide each side by the coefficient. When the coefficient of the variable is a fraction, you will instead multiply both sides by the recip- rocal of the coefficient. Technically, you could still divide both sides by the coefficient, but that involves division of fractions which can be trickier. Example ᎏ 2 3 ᎏ m + ᎏ 1 2 ᎏ =12 – ᎏ 1 2 ᎏ =– ᎏ 1 2 ᎏ ᎏ 2 3 ᎏ m =11 ᎏ 1 2 ᎏ ᎏ 3 2 ᎏ • ᎏ 2 3 ᎏ m =11 ᎏ 1 2 ᎏ • ᎏ 3 2 ᎏ ᎏ 3 2 ᎏ • ᎏ 2 3 ᎏ m = ᎏ 2 2 3 ᎏ • ᎏ 3 2 ᎏ m = ᎏ 6 4 9 ᎏ Undo the addition of ᎏ 1 2 ᎏ by subtracting ᎏ 1 2 ᎏ from both sides of the equation. Multiply both sides by the recip- rocal of the coefficient. Convert the 11 ᎏ 1 2 ᎏ to an improper fraction to facilitate multiplication. The variable m is now isolated on the left side of the equation, and its value is ᎏ 6 4 9 ᎏ . Equations with More than One Variable Equations can have more than one variable. Each variable represents a different value, although it is possible that the variables have the same value. Remember that like terms have the same variable and exponent. All of the rules for working with variables apply in equations that contain more than one variable, but you must remember not to combine terms that are not alike. Equations with more than one variable cannot be “solved,”because if there is more than one variable in an equa- tion there is usually an infinite number of values for the variables that would make the equation true. Instead, we are often required to “solve for a variable,” which instead means to isolate that variable on one side of the equation. Example Solve for y in the equation 2x + 3y = 5. There are an infinite number of values for x and y that that satisfy the equation. Instead, we are asked to isolate y on one side of the equation. 2x + 3y =5 – 2x =– 2x ᎏ 3 3 y ᎏ = ᎏ –2x 3 +5 ᎏ y = ᎏ –2x 3 +5 ᎏ – THEA MATH REVIEW– 123 Cross Multiplying Since algebra uses percents and proportions, it is necessary to learn how to cross multiply. You can solve an equa- tion that sets one fraction equal to another by cross multiplication. Cross multiplication involves setting the cross products of opposite pairs of terms equal to each other. Example ᎏ 1 x 0 ᎏ = ᎏ 1 7 0 0 0 ᎏ 100x = 700 ᎏ 1 1 0 0 0 0 x ᎏ = ᎏ 7 1 0 0 0 0 ᎏ x =7 Algebraic Fractions Working with algebraic fractions is very similar to working with fractions in arithmetic. The difference is that alge- braic fractions contain algebraic expressions in the numerator and/or denominator. Example A hotel currently has only one-fifth of their rooms available. If x represents the total number of rooms in the hotel, find an expression for the number of rooms that will be available if another tenth of the total rooms are reserved. Since x represents the total number of rooms, ᎏ 5 x ᎏ (or ᎏ 1 5 ᎏ x) represents the number of available rooms. One tenth of the total rooms in the hotel would be represented by the fraction ᎏ 1 x 0 ᎏ . To find the new number of available rooms, find the difference: ᎏ 5 x ᎏ – ᎏ 1 x 0 ᎏ . Write ᎏ 5 x ᎏ – ᎏ 1 x 0 ᎏ as a single fraction. Just like in arithmetic, the first step is to find the LCD of 5 and 10, which is 10. Then change each fraction into an equivalent fraction that has 10 as a denominator. ᎏ 5 x ᎏ – ᎏ 1 x 0 ᎏ = ᎏ 5 x( ( 2 2 ) ) ᎏ – ᎏ 1 x 0 ᎏ = ᎏ 1 2 0 x ᎏ – ᎏ 1 x 0 ᎏ = ᎏ 1 x 0 ᎏ Therefore, ᎏ 1 x 0 ᎏ rooms will be available after another tenth of the rooms are reserved. – THEA MATH REVIEW– 124 Reciprocal Rules There are special rules for the sum and difference of reciprocals. The reciprocal of 3 is ᎏ 1 3 ᎏ and the reciprocal of x is ᎏ 1 x ᎏ . ■ If x and y are not 0, then ᎏ 1 x ᎏ + ᎏ 1 y ᎏ = ᎏ x y y ᎏ + ᎏ x x y ᎏ = ᎏ y x + y x ᎏ . ■ If x and y are not 0, then ᎏ 1 x ᎏ – ᎏ 1 y ᎏ = ᎏ x y y ᎏ – ᎏ x x y ᎏ = ᎏ y x – y x ᎏ . Translating Words into Numbers The most important skill needed for word problems is being able to translate words into mathematical operations. The following will be helpful in achieving this goal by providing common examples of English phrases and their mathematical equivalents. Phrases meaning addition: increased by; sum of; more than; exceeds by. Examples A number increased by five: x + 5. The sum of two numbers: x + y. Ten more than a number: x + 10. Phrases meaning subtraction: decreased by; difference of; less than; diminished by. Examples 10 less than a number: x – 10. The difference of two numbers: x – y. Phrases meaning multiplication: times; times the sum/difference; product; of. Examples Three times a number: 3x. Twenty percent of 50: 20% ϫ 50. Five times the sum of a number and three: 5(x + 3). Phrases meaning “equals”: is; result is. Examples 15 is 14 plus 1: 15 = 14 + 1. 10 more than 2 times a number is 15: 2x + 10 = 15. – THEA MATH REVIEW– 125 . involves setting the cross products of opposite pairs of terms equal to each other. Example ᎏ 1 x 0 ᎏ = ᎏ 1 7 0 0 0 ᎏ 100x = 70 0 ᎏ 1 1 0 0 0 0 x ᎏ = ᎏ 7 1 0 0 0 0 ᎏ x =7 Algebraic Fractions Working. minutes of calls, with a charge of $.55 for each minute thereafter. Write an algebraic expression for the cost of a month’s mobile phone bill: $39.99 + $.55x,where x represents the number of additional. degree of a polynomial is the largest sum of the terms’ exponents. Examples ■ The degree of the trinomial x 2 + 2x – 1 is 2, because the x 2 term has the highest exponent of 2. ■ The degree of the