Algebra 1 Final Exam PDF

Name School Section MATH Algebra 1 R...

Name School Section MATH Algebra 1 Reviewer A. NUMBER SYSTEM The number 0.333…is a repeating and non- D. GREATEST COMMON FACTOR (GCF) A number is an item that describes a magnitude terminating decimal. As a rule, a non- A factor is a number that divides into a larger or a position. Numbers are classified into two terminating but repeating (or periodic) decimal number evenly. The greatest common factor types, namely cardinal numbers and ordinal is always a rational number. Also, all integers (GCF) is the largest number that divides into two numbers. are rational numbers. or more numbers evenly. Greatest common factor is the same as greatest common divisor (GCD). Cardinal numbers are numbers which allow us to 4. Irrational numbers – are numbers which cannot count the objects or ideas in a given collection. be expressed as a quotient of two Past ECE Board Exam: Example, 1,2,3...,1000, 100000 while ordinal integers. What is the GCF of 70 and 112? numbers state the position of the individual objects in a sequence. Example, First, second, E. EXPONENTS Examples: 2 , π, e,... third... Exponent is a number that gives the power to The numbers in the examples above can which a base is raised. For example, in 32, the A system of numbers is a diagram or chart which base is 3 and the exponent is 2. shows the two sub-classifications of the two basic never be expressed exactly as a quotient of classifications of numbers, namely real numbers two integers. They are in fact, a non- terminating number with non-terminating Exponent should not be misunderstood as and imaginary numbers. “power” Power is a word that is almost never used decimal. in its correct, original sense any more. Strictly System of Past ECE Board Exam: speaking, if we write 32 = 9, then 3 is the base, Numbers The number 7 + 0i is a/an _______ number. 2 is the exponent and 9 is the power. But almost A. irrational C. real * everyone, including most mathematicians, would B. imaginary D. complex say that 3 is the power and that “power” and “exponent” mean the same thing. The misuse has Real Imaginary Past ECE Board Exam: probably come from a misunderstanding of Numbers Number The number 0.123123123… is a _____ number. statements such “nine is the second power of A. irrational C. rational * three”. B. imaginary D. complex The exponential notation states that if a is a real Irrational Rational A complex number is an expression of both real number, variable or algebraic expression and n is numbers numbers and imaginary number combined. It takes the form a positive number, then: of a + bi, where “a” and “b” are real numbers. an = a ⋅ a ⋅ a ⋅ a ⋅
Integers If a = 0, then pure imaginary number is produced n factors while real number is obtained when b = 0. Property Example Natural Negative An imaginary number is denoted as “i” which is Zero equal to the square root of negative one. In some 1. am + an = am +n x 2 + x3 = x 2 + 3 = x5 numbers numbers other areas in mathematical computation, especially in electronics and electrical engineering am x8 2. = am −n = x8 −3 = x5 The number system is divided into two categories it is denoted as “j”. Imaginary number and its an x3 namely, real numbers and imaginary number. equivalent: n 2 Real numbers are classified as follows: i = −1 i3 = -i = - −1 3. (a ) m = amn (y ) 6 = y12 i2 = -1 i4 = 1 1. Natural numbers – numbers which are 4. ( ab )m = ambm ( 2x )4 = 24 x 4 = 16x 4 Past ECE Board Exam: considered as the “counting 5 m 4 numbers”. Find the value of (1 + i ). a am 2 24 16 5.   = m   = 4 = 4 b b x x x Examples: 1,2, 3… B. SIGNIFICANT DIGITS m 5 2. Integers – are all the natural number, the Significant figures or digits are digits that define 6. a n = am n ( 4x ) 3 = 3 ( 4x )5 negative of the natural numbers the numerical value of a number. A digit is and the number zero. considered significant unless it is used to place a 1 1 decimal point. 7. a−m = m x −5 = a x5 Examples: - 4, -1, 0, 3, 8 The significant digit of a number begins with the 0 3. Rational numbers – are numbers which can be first non-zero digit and ends with the final digit, 8. a0 = 1 ( a ≠ 0 ) (x 2 +2 ) =1 expressed as a quotient (ratio) of whether zero or non-zero. Past ECE Board Exam: two integers. The term “rational” comes from the word “ratio”. Past ECE Board Exam: 17 Solve for x. x 2 / 3 + x −2 / 3 =. 2 The number 0.004212 has how many significant 4 Examples: 0.5, , -3, 0.333... digits? Past ECE Board Exam: 3 In the above example, 0.5 can be expressed Solve for x. 4 x + 2x − 30 = 0 C. LEAST COMMON MULTIPLE (LCM) 1 −6 as and –3 can be expressed as , A common multiple is a number that two other Past ECE Board Exam: 2 2 numbers will divide into evenly. The least common hence the two examples are rational numbers. −5 / 2 multiple (LCM) is the lowest multiple of two 5(4x −1)  1  Solve for x. 62 =5  numbers.  36  The number 0.333... can also be expressed as 1 Past ECE Board Exam: and therefore a rational number. F. LOGARITHMS 3 What is the least common multiple of 15 and 18? The logarithm of a number or variable x to base b, logb x , is the exponent of b needed to give x.  Past ECE Board Exam: ( x + y )0 = 1 What is the value of k if ( x + 4 ) is a factor of x3 + 1 ( x + y) = x + y Log 2 16 = 4 may be written as 24 = 16 2x2 – 7x + k ? 2 ( x + y ) = x 2 + 2xy + y 2 ( x + y )3 = x3 + 3x 2 y + 3xy 2 + y 3 I. QUADRATIC EQUATION The term “logarithm” comes from Greek words, Quadratic is an expression or an equation that  “logus” meaning “ratio” and “arithmus” meaning contains the variable squared, but not raised to As observed in the binomial expansions above, “number”. John Napier (1550 – 1617) invented any higher power. Quadratic equation in x some properties were established and are logarithm in 1614 using e = 2.718… for its base. contains x2 but not x3. A quadratic equation in x is enumerated as follows: In 1616, through the suggestion of John Napier, also known as a second-degree polynomial Henry Briggs improved the logarithm using 10 as equation. Properties of Binomial Expansion of (x + y)n: the base. The logarithm with base 10 is known as common logarithm or the Briggsian logarithm. The general quadratic equation is expressed as: 1. The number of terms in the resulting expansion is equal to n + 1. 2. The exponent of x decreases by 1 in The natural logarithm can be converted into a Ax 2 + Bx + C = 0 common logarithm and vice versa. To obtain this, succeeding terms, while that exponent of y a factor known as the modulus of logarithm is increases by 1 in succeeding terms. where, A, B and C are real numbers and with A 3. The sum of the exponents of each term is necessary, such as: ± 0. When B = 0, quadratic equation is known as equal to n. a pure quadratic equation. 4. The first term is xn and the last term is yn and log x = 0.4343ln x ln x = 2.3026log x each of the terms has a coefficient of 1 The quadratic formula: 5. The coefficient increases and then decreases The coefficients 0.4343 and 2.3026 are the in a symmetric pattern. referred to as the modulus of logarithm. −B ± B 2 − 4AC x= The Pascal’s Triangle: Properties of Logarithms: 2A Each number in the triangle is equal to the sum of the two numbers immediately above it. 1. log ( xy ) = log x + log y The quantity B 2 − 4AC in the above equation is known as the discriminant. The discriminant Binomial Pascal’s Tiangle x determines the nature of the roots of the quadratic (x + y)0 1 2. log   = logx − log y y equation. (x + y)1 1 1 (x + y)2 1 2 1 B 2 − 4AC Nature of roots (x + y)3 1 3 3 1 3. log xn = nlog x 0 Only one root (x + y)4 1 4 6 4 1 log x (Real and equal) (x + y)5 1 5 10 10 5 1 4. logb x = (x + y)6 1 6 15 20 15 6 1 logb >0 Real and unequal