MTH 110-College Algebra Chapter R-Basic Concepts of Algebra R.1 I. Real Number System Please indicate if each of these numbers is a W (Whole number), R (Real number), Z (Integer), I (Irrational number), Q (Rational number), N (Natural number). Examples: 6, 7, 5. 4, 3 7, 3 8, 6. 444, 6. 4444..., 0. 000000... *: symbol for element of, indicating that it is a member belonging to a set. Example: 0. 33Q * : symbol for subset of;when all the elements of one set are elements of a second set, we say that the first set is a subset of the second set. Example: Q R, I R, Z R, W Z, Z Z, N W, Zero,0 W 1
II. Properties of Real Numbers For any real numbers a, b, and c: a + b = b + a commutative property of addition a b = b a commutative property of multiplication a + ( b + c ) = ( a + b ) + c associative property of addition a ( b c ) = ( a b ) c associative property of multiplication a + 0 = 0 + a = a additive identity property a -a = a + ( - a ) = 0 additive inverse property a 1 = 1 a = a multiplicative identity property a 1 a = 1 such that a 0 multiplicative inverse property a ( b + c ) = a b + a c distributive property Example: Indicate which is the appropriate property 1. 4 6 6 4. 6 ( x y) ( 6 x) y 3. 0 ( 0) 0 4. 9 1 19 9 5. 4( x y) 4x 4y
III. Absolute Values, For any real number a, a { a a 0 a a 0 Example: 1. 6. 3 5 3. 8 4. 4 ( ) 5. 3 = 6. 3 5 7. 81. 8. 4 IV. Distance Absolute value of the difference between two numbers Examples: 1. distance between - and -8. distance between - 4 and -14 3. distance between 1.1 and 5.1 4. distance between 15 8 and 3 1 5. distance between 16 and -8 3
V. Intervals: Types, Notation, and Graphs Type Interval Notation Set Notation Graph Open Closed Half-open Half-open (a, b ) { x a x b} [a, b] { x a x b} [a, b ) { x a x b} (a, b ] { x a x b} Open ( a, ) { x x a} Half-open [a, ) { x x a} Open (, b ) { x x b} Half-open (, b ] { x x b} *Two possible ways of graphing will be demonstrated in class. Write interval notation for each set and graph the set: Examples: 1. { x 4 x. 5}. { x 4 x 10} 3. { x x. 6} 4. { x x 1 } 4
R. I. Integers as Exponents A. For any positive integer n, a n a a a a a n times such that a is the base and n is the exponent Example: a 3 a a a 3 3 3 3 3 = 7 B. For any nonzero real number a and any integer n, a 0 1 and a n 1 a n Examples: a. 6 0 1. ( ) 0. ( ) 3 3. 3 = 4. 4 = 5
II. Properties of Exponents a a a m n m n a a m n a ( mn) such that a 0 ( a ) ( ab) a m n mn a b m m m a ( ) b m m a such that b 0 m b Examples: 1. ( x ) 5. 45x 15x 8 3. y 5 y 5 4. x x 5. y y 6 3 6. y y 4 7. ( x 3 ) 4 8. y y 3 9. m 5 5 m 10. ( x 3 ) 5 10 8 7 4 4a b c 5 7x y 11. ( ) = 1. ( ) 6 3 5 8 3a b c 9x z 3 6
III. Scientific Notations Scientific notation for a number is an expression of the type N x 10 m, such that1 N 10, N is in decimal notation, m is an integer In scientific notation, numbers appear as a number greater than or equal to 1 and less than 10 multiplied by some power of 10. Ex. 1. FM radio signal may be 14,00,000,000 hertz (cycles per second); in scientific notation, this is 1.4 10 10 hertz.. Diameter of an atom is 0.0000000001 meter In scientific notation, this is110 10 meter Try to express the result in scientific notation: 1. ( 91. 10 17 )( 8. 10 3 ). 6. 4 10 8. 0 10 7 6 = 3. 145, 000, 000 = 4. 0. 00876 = Try to express the following without exponents: 1.. 3 10 4 = 7
. 8. 97 10 5 = 3. 146. 10 6 = 4. 4. 57 10 3 = *In 000, the number of people living in the world was about 6. 09 10 9. The number of people lived in US at that time was about. 74 10 8. How many people lived outside of US in 000? *We have proof that there are at least 1 sextillion, 10 1, stars in the Milky Way. Write this number without the use of exponent. 8
Examples: 1. Convert each of the followings to decimal notation: a. 7. 654 10 5 = b. 6. 4510 5. Convert each of the followings to scientific notation: a. 876, 40, 000 b. 0. 00054 c. ( 91. 10 17 )( 8. 10 3 ) d. 6. 4 10 8. 0 10 7 6 = e. 4. 3 ( 34. 10 1 ) f.. 5 10 510 6 g. 4. 6. 6 10. 5 4 10 8 9
IV. Order of Operations Please Excuse My Dear Aunt Sally ( ) a exp + - 1. Exponential expression and calculations within grouping symbols first and always from left to right.. Multiplication and division from left to right 3. Addition and subtraction from left to right Examples: 1. ( 6 3) 3. 8( 5 3) 10 = 3. 4{ 3a [ 4a ( b a )]} = 4. [ ( ) 4 8 6 4 ]( 3 8 ) 3 ( 5) 5. b 3[ 5b b( b)] = 6. 4( 8 6) 4 3 8 1 0 3 19 = 10
V. Compound Interest Formula Principal P is invested at an interest rate i, compounded n times per year, in t years it will grow to an amount A given i nt r by: A P( 1 ) or A P( 1 ) n n Examples: A: total amount, P: principal, i r nt annual interest rate n: number of times compounded per year, t: number of years 1. Suppose $9550 is invested at 5.4%, compounded semiannually. How much is in the account at the end of 7 years?. Suppose $6700 is invested at 4.5%, compounded quarterly. How much is in the account at the end of 6 years? 11
3. Suppose you will be needing $0,000 in ten years, how much do you need to invest now (principal), if the investment will be earning interest at 10% and compounding semiannually? 4. Suppose you will be needing $0,000 in ten years, how much do you need to invest now {principal}, if the investment will be earning interest at 10% and compounding quarterly. 5. Suppose you will be needing $0,000 in ten years, how much do you need to invest now (principal), if the investment will be earning interest at 0% and compounding semiannually. 1
6. The interest earned on an $800 investment at 7 1 % annual interest compounded monthly 4 for 6 months is? 7. Shane begins a new job with an annual salary of $40,000 and a guarantee of a % salary increase every year for the first 5 years. After that, he is given 3% increase each year. What will be his salary in 8 years? 8. A couple want to have $100,000 in 1 years for their newborn son. How much money should be deposited now in an account earning 6 1 % annual interest compounded monthly to achieve this goal? 13
R.3 I. Polynomials in One Variable n n 1 a x a x... a x a x a n n 1 1 0 and that n is a nonnegative integer, and a n,..., a 0 are real numbers called coefficients and a n 0 A. Definition 1. Terms. Degree of the polynomial 3. Leading coefficient 4. Constant term 5. Descending order Examples: 4 3 1 1. x 8x x 0. y 6y 3 14
6. Monomial 7. Binomial 8. Trinomial II. Polynomial in Several Variables A. Definition 1. Degree of a term - sum of the exponents of the variables in that term.. Degree of a polynomial - the degree of the term of highest degree. Examples: 1. 9ab 3 1a b 4 9. 7x 4 y 3 5x 3 y 3x y 6 15
III. Expressions That Are Not Polynomials 5 a. x 5x b. 0 x c. x y 3 y 1 7 IV. Addition and Subtraction of Polynomials Like terms - terms/expressions that have same variables raised to the same powers. Combine/collect like terms Examples: 3 3 1. ( 5x 3x x) ( 1x 7x 3). ( 8x y 3 9xy) ( 6x y 3 3xy) 3. ( 3x x x 3 ) ( 5x 8x x 3 4) 16
V. Multiplication of Polynomials A. (binomial)(binomial) : binomial multiplied by binomial, use FOIL ( x 4)( x 3 ) ( x x) ( x 3) ( 4 x) ( 4 3) F O I L x 3x 4x 1 = x 7x 1 Examples: 1. ( x 5)( x 3). ( a 3)( a 5) 3. ( x 3y)( x 5y) = 4. ( 4x 1) ( 4x 1)( 4x 1) 5. ( 5x 1) ( 5x 1)( 5x 1) 17
6. ( 3y ) ( 3y )( 3y ) 7. ( a 1) ( a 1)( a 1) 8. ( a 1)( a 1) 9. ( 3y )( 3y ) 10. ( 5x 1)( 5x 1) B. Special Products of Binomials 1. ( A B) A AB B. ( A B) A AB B 3. ( A B)( A B) A B Examples: 1. ( 3x ). ( 3x ) 3. ( 3x )( 3x ) 4. ( 5x 4)( 5x 4) 5. ( a n b n )( a n b n ) 18
C. Multiplying Two Polynomials Examples: 1. ( a b)( a 3 ab 3b ). ( 4x 4 y 7x y 3y)( y 3x y) 3. ( x 3y 4)( x y) 19
R.4 I. Factoring Terms with Common Factors Examples: 1. 16 1x 4x = 4( 4 3x x ) 3. 14x y 35x y = 3 3. 1x y 4x y = II. Factoring by Grouping Pairs of terms have a common factor that can be removed in a process called factoring by grouping. *Hint: usually for 4 terms Examples: 3 1. x 3x 5x 15 = x ( x 3) 5( x 3) =( x 3)( x 5) 3. p p 9 p 18 = 3 3. y 3y 4y 1 = 4a a a 4. 4x 1x 10x 30 = 5. x ax bx ab = 0
III.Factoring Trinomials Some trinomials can be factored into the product of two binomials. A. To factor a trinomial of the form x bx c, we look for two numbers with a product of c and a sum of b. Examples: 1. z z 4 = ( z 6)( z 4). x x 0 = 3. x 18x 81= 4. y 4y 1= 5. x y 18xy 64 = 6. x 5xy 6y = 7. 10 3x x = (hint: factoring -1 from the trinomial) 8. a 8ab 33b = 1
B. Factoring a trinomial of the form ax bx c, a 1 Use the trial factor method: use the factor of a and the factors of c to write all of the possible binomial factors of the trinomial. Then use FOIL to determine the correct factorization. To reduce the number of trial factors that must be considered, remember the following: 1. Use the sign of the constant term and the coefficient of x in the trinomial to determine the signs of the binomial factors. If the constant term is positive, signs of the binomial factors will be the same as the sign of the coefficient of x in the trinomial. If the sign of the constant term is negative, the constant terms in the binomials will have opposite signs.. If the terms of the trinomial do not have a common factor, then the terms in either one of the binomial
factors will not have a common factor. Examples: a. Factor: 3x 8x 4, + factors of 3 - factors of 4 The terms have no common (coeff. of x ) (constant term) factor. 1, 3-1, -4 The constant term is positive. -, - Coefficient of x is negative. The binomial constant will be negative. Write trial factors. Use the Trial Factors Middle Term Outer and Inner products of ( x 1)( 3x 4) 4x 3x 7x FOIL to determine the middle ( x 4)( 3x 1) x 1x 13x term of the trinomial. ( x )( 3x ) x 6x 8x Write the trinomial in factored form3x 8x 4 ( x )( 3x ) 3
b. Factor: 10 x x + factor of 10 Factors of - The terms have no common (constant term) (coeff. of x ) factor. The coefficient of x is -. 1, 10 1, - Thus the signs of the binomi-, 5-1, als will be opposites. Write trial factors. Use the Trial Factors Middle Term Outer and Inner products of ( 1 x)( 10 x) x 10x 8x FOIL to determine the middle ( 1 x)( 10 x) x 10x 8x term of the trinomial. ( x)( 5 x) 4x 5x x ( x)( 5 x) 4x 5x x Write the trinomial in factored form:10 x x ( x)( 5 x) 3 c. Factor: 10y 44y 30y, the GCF is y. Factor the GCF from the terms: 10y 3 44y 30y y( 5y y 15) 4
Factor the trinomial + Factors of 5 Factors of -15 The constant term is ( coeff.. of y ) (constant term) negative. 1, 5-1, 15 The binomial constants will 1, -15 have opposite signs. -3, 5 3, -5 Write trial factors. Use the Trial Factors Middle Term Outer and Inner products of ( y 1)( 5y 15) 15y 15y 0 FOIL to determine the middle ( y 15)( 5y 1) y 75y 74y term of the trinomial. ( y 1)( 5y 15) 15y 5y 10y ( y 15)( 5y 1) y 75y 74y ( y 3)( 5y 5) 5y 15y 10y ( y 5)( 5y 3) 3y 5y y ( y 3)( 5y 5) 5y 15y 10y ( y 5)( 5y 3) 3y 5y y Write the trinomial in factored form 10y 44y 30y y( y 5)( 5y 3) 5
C. Trinomial of the ax bx x form can also be factored by grouping. To factor ax bx c, first find two factors a c whose sum is b. Then use factoring by grouping to write the factorization of the trinomial. Examples: a. Factor: 3x 11x 8 Find two positive factors of 4 + Factors of 4 Sum (ac 3 8) whose sum is 11, the 1, 4 5 coefficient of x., 1 14 3, 8 11 The required sum has been found. The remaining factor need not be checked. Use the factors of 4 3x 11x 8 3x 3x 8x 8 whose sum is 11 to write ( 3x 3x) ( 8x 8) 11x as 3x 8x. = 3x( x 1) 8( x 1) Factor by grouping. ( x 1)( 3x 8) Check: ( x 1)( 3x 8) 3x 8x 3x 8 3x 11x 8 6
b. Factor: 4z 17z 1 Find two factors of -84 [ac 4 ( 1)] whose sum is - 17, the coefficient of z.. Factors of -84 Sum 1, - 84-83 - 1, 84 83, - 4-40 -, 4 40 3, - 8-5 -- 3, 8 5 4, - 1-17 (Once the required sum is found, the remaining factors need not be checked). Use the factors of - 84 whose sum is -a7 to write - 17z as 4z 1z. Factor by grouping. Recall that 1z 1 ( 1z 1) 4z 17z 1 4z 4z 1z 1 ( 4z 4z) ( 1z 1) 4z( z 1) 1( z 1) ( z 1)( 4z 1) 7
Examples: 1. Factor: 6x 11x 10. Factor: 1x 3x 5 3. Factor: 30y xy 4x y 4. Factor: 4x 15x 4 5. Factor: y 18y 7 6. Factor: 5 4x x 7. Factor: 4a a 5 8. Factor: y 5 8y 3 15y 8
D. Special Factorizations 1. Factors of the Difference of Two Perfect Squares A B ( A B)( A B) Examples: 1. x 5 ( x 5)( x 5). 9x 5 3. 4x 81y = 4. 5x 1 = 5. x 36y 4. Factors of a Perfect Square Trinomial A AB B ( A B) A AB B ( A B) Examples: 1. x 8x 16. 9x 1x 4 3. 4x 0x 5 9
3. To Factor the sum or the difference of two cubes 3 3 A B ( A B)( A AB B ) 3 3 A B ( A B)( A AB B ) Examples: 1. a 64y 3 a ( 4y) 3 ( a 4y)( a 4ay 16y ) 4. 64y 15y 3. 8x 3 y 3 z 3 4. x 3 y 3 1 3 3 5. ( x y) x 30
R.5 A rational expression is the quotient of two polynomials. For example, 3,, 5 x 3 y y 3 4x 5 are rational expressions I. Domain of a Rational Expression Any number that makes the denominator zero is not in the domain of a rational expression. Examples: Find the domain of the following 1. 5 x 5. x 4 3. x 4x 5 3 t 4 *For a, use set-builder notation : {x x is a real number and x 5} 4. x 3 5x 5x 6 5. x 1 6. x 1 5x 3x 9 31
II. To simplify rational expressions, use the fact that: a c b c a c a 1 b c b a b Examples: Simplify 1. 3 6x 9x 1x 18x =. 6x 4x 1x 48x 4 3 3 = 3. 1 x y 6 x y 6x y 3 3 3 = 4. 0x 15x 3 15x 5x 0x = *The simplified version may also be considered as the equivalent expression of the original expression. ex. x x x 6 and 1 x 3 are equivalent expressions. 3
III.Multiply or divide and simplify each of the following. Examples: 1. x x 4 x 9 3 x x =. x 6x 3x 6 6x 1 8x 1x 3 = 3. 6x y 35a b 4 5 3 3 1x y = 4 5 7a b 4. 4x 4y 6x y 3x 3xy x y xy = 33
IV. Adding and Subtracting Rational Expressions When adding or subtracting rational expressions, it is often necessary to express the rational expressions in terms of a common denominator. This common denominator is the least common multiple (LCM) of the denominators. Examples: 1. a 3 a a a 1 a a 6 = The LCM of ( a a) and ( a a 6) : a a a( a ); a a 6 ( a 3)( a ) LCM is: a( a )( a 3) a 3 a a a 3 a 3 4a 9 a a a 3 ( ) a( a )( a 3) a 1 a a a a a a 1 a a a 6 6 a( a )( a 3) Now they have the LCM, add: 4a 9 a a a( a )( a 3) a( a )( a 3) = 5a a 9 a( a )( a 3) 34
. 3x x 11x 15 x x 3x = 3. x 5 x 11x 30 x 9x 0 4. y y y 0 y 4 5. x y x 3y 3y x 6. a 3 a 5a a a 9 5 7. 5 5x x 5 6x 35
V. Complex Rational Expressions A. Definition A complex rational expression has rational expressions in its numerator or denominator or both. B. Simplifying complex rational expressions Method 1: Multiply the numerator and denominator by the LCD of all of the denominators. Method : Take care of the operations in the numerator and the denominator, then the numerator multiply by the reciprocal of the denominator. Examples: 1. Method 1: 1 1 x y 1 1 x y 1 1 x y 1 1 x y xy xy 36
Method : 1 1 x y 1 1 x y y x xy y x xy y x xy xy y x. 1 1 a b 1 1 a b 3 3 3. 5 1 4. 5 a a 5 a = 37
R.6. Radical Notation and Rational Expressions I. Radical Notation A number c is said to be an nth root of a if c n a. Square root: if n Cube root: if n 3 If given n a, the symbol n is called a radical, n is the index, and a is the radicand. The positive root is called the principal root. To denote a negative root, use 4, 8, etc. Examples: 36 6, 36 6, 5 3 43 ( ) 3 5 3 4 8, 16 is not a real number n * When a is negative and n is even, a is not a real number. 38
II. Simplifying Radical Expressions If a and b are real numbers or expressions for which the given roots exist. m, n are natural numbers. Here are some properties of radicals. n n A. If n is even, a a n n B. If n is odd, a a C. n a n b n ab D. n a b n a ( b 0 ) n b m E. a ( a ) n n m Examples: Simplify 5 1. 3. 18c d 4 = 3. 6 1 4 m n 64 4. 3 3 40m 5m 5. 3 7x 9 6. 9a 4 b 8 = 39
III. Rationalizing Denominators or Numerators Removing the radicals in a denominator or numerator. It is done by multiplying by 1 in such a way so to obtain a perfect nth power. Examples: 1. rationalizing the denominator 3 3 6 6 4 4 6. rationalizing the numerator x y x y x y ( x ) ( y) x y 5 5 x y 5 x 5 y 5 x 5 y 3. Rationalizing the numerator x 3 y = 4. Rationalizing the denominator 3 7 9 3 40
IV. Rational Exponents For any real number a and any natural numbers m and n, a 1/ n n m/ n n a a a m a m/ n 1 1 m/ n n a a m Examples:Convert exponent expressions to radical notations 3/ 4 4 3 a. 7 7 b. 8 5/ 3 = c a 3 5 / = Examples: Convert to exponential notation and simplify. 1. ( 4 7xy ) 5 ( 7xy) 5/ 4. 6 x 3 3 3. 7 4 4. ( 13) 5 5. 5 3 3 4 = 6. 7 = 41
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