P.1 Algebraic Expressions, Mathematical models, and Real numbers. Exponential notation: Definitions of Sets: A B. Sets and subsets of real numbers:

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1 P.1 Algebraic Expressions, Mathematical models, and Real numbers If n is a counting number (1, 2, 3, 4,..) then Exponential notation: b n = b b b... b, where n is the Exponent or Power, and b is the base Definitions of Sets: Union written A B is the members of set A or set B or both sets (combine the two sets) Intersection written A B is the members that are in both A and B (overlap of the two sets) null set is the empty set and is written { } or Example: Find the a) union, and b) intersection A = { 7, 8, 9, 10, 11}, B = { 6, 8, 10, 12} Be sure to use the notation: a) union = b) intersection Sets and subsets of real numbers: Rational numbers (written can be expressed as a fraction 17 = 17, = 0.4, 2 3 = = 0.6 Integers (written Q ) numbers that are finite or repeat, or Z ) all positive and negative numbers without decimals (including zero)... 3, 2, 1, 0, 1, 2, 3,... Irrational numbers (written I ) numbers whose decimals are not finite nor repeating 2, 3, π, e, 2 π Whole numbers (written W ) positive numbers (and zero) 0, 1, 2, 3,... Natural numbers (written N ) positive numbers (without zero) 1, 2, 3 4, x = { x if x 0 x if x < 0 Absolute Value In other words, the absolute value changes the interior expression to a positive number! Note: a b = b a = / a b Absolute values do not distribute across subtraction

2 P.2 Exponents and Scientific Notation Simplifying Exponential Expressions: 1. If necessary remove parenthesis by distributing powers to powers within (or if adding/subtracting, writing the expression multiple times and distributing) ( xy 2) 3 = ( 4a) 3 = 2. If necessary simplify powers of powers using the power rule ( x 4 ) 0 = 3. If necessary be sure that each base appears only once in a product or quotient by using the Products and Quotients rule (note these don t apply across addition or subtraction) x 4 x 3 = 4. Rewrite exponents with zero, negative, or 1 powers x 5 = x 8 The most common error seen is ( 4x 3) 2 = 16x 2 9. What is wrong? Scientific Notation: A number is written in scientific notation when it is written in the form a 10 n where 1 a < 10 and n is an integer. To convert from scientific notation to decimal notation: The decimal moves to the right if n is positive, and to the left if n is negative. To convert from a decimal to scientific notation (writing the number in in the form a Determine a the numerical factor. Move the decimal in the given number to obtain an a between 1 and n is the number of decimal places moved. a. n is positive if the decimal was moved to the left b. n is negative if the decimal was moved to the right n Note: If the result of an operation changes a to be 10 or above, or less than 1, the power must be adjusted to put it back in scientific notation. E or EE in your calculator is understood to be 10^

3 P.3 Radicals and Rational Exponents Principal square root of a If a is nonnegative then if b 2 = a then b = a = a 1/2 nth roots: n n a = b means b = a if n is odd if n is even n a n = a (signs are the same) n a n = a (sign is always positive) n a = a 1/n aka an 1 = n a where n 2 is an integer. Exponent rules can still be applied. if n is even and a is negative then the result is an imaginary number. a m n = n a m = ( n a) m When solving an equation if b 2 = 25, or when b 2 = 5 unless b must be positive, there are two solutions b = + 25 = 5 and b = 25 = 5 which can be abbreviated b = ± 25 = ± 5 Combining Radicals: 1. simplify the radicals 2. the radical s root and the interior of the radicals must match 3. Add the numbers out front 4. write the radical and the same interior beside it (adding like terms only combine numbers out front) Rationalizing the denominator: 3 scenarios: Multiply by the radical on the denominator (only one term) or the conjugate of the denominator (multiple terms change sign ) One term on the denominator: Two terms on the denominator (square root) Two terms on the denominator (cube root)

4 Exponents can be written in rational form if needed before rules are used. P.4 Polynomials Degree of a basic (monomial) ax n is n. constant (term without variables) is 0 polynomial is the greatest degree of all of the terms (monomials). The leading coefficient is the number in front of the variable of the monomial with the greatest power Adding and subtracting polynomials 1. Distribute the subtraction (if applicable) 2. Combine like terms (same variables, same powers) Multiplying Monomials: multiply the numbers out front, add powers of like bases ( 8x 6 )5x 3 = ( 8 5)x 6+3 = 40x 9 Multiplying polynomials: Monomial by polynomial Binomial by binomial (double distribute/foil) Polynomial by polynomial multiple distributes, then combine like terms (see below) Squaring a binomial: (middle term doubles same sign, Special products: exterior terms square positive sign), Multiplying binomials of different signs (middle terms eliminate, square first and last terms, last term negative!) Cubing a binomial cube first and last terms left middle term = 3 first 2 l ast right middle terms = 2 3 f irst last Note: power of left term reduces left right & power of right term reduces right left

5 P.5 Factoring Polynomials: GCF By Grouping Both (divide each term by the GCF Trinomials (with leading coefficients) Trinomials (without leading coefficients) Note: the middle two terms must add up to the middle term multiply to the product of the first and last! Find two numbers that: multiply to the last term add to the middle term Signs shortcut 1. If the last term is NEGATIVE the signs in the binomials are different larger number has the same sign as the middle term in the trinomial 2. If the last term is POSITIVE the signs in the binomial are the same they have the same sign as the middle term in the trinomial Special Cases: Sum/Difference of two perfect Cubes: Difference of Two Perfect Squares:

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