Copy in your notebook: Add an example of each term with the symbols used in algebra 2 if there are any.


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1 Algebra 2  Chapter Prerequisites Vocabulary Copy in your notebook: Add an example of each term with the symbols used in algebra 2 if there are any. P1 p counting(natural) numbers  {1,2,3,4,...} 2. whole numbers  {0,1,2,3...} 3. integers  {...3, 2,1,0,1,2,3...} p rational numbers  { a/b a and b are integers with b 0}  numbers that can be written as fractions. 5. irrational numbers  nonterminating, nonrepeating decimals. p real numbers  rational and irrational numbers p number line  a line with units marked off representing the integers and the spaces between representing rational and irrational numbers  a representation of the real numbers. 8. coordinates  numbers corresponding to points on the number line. p is a member of is not a member of is a subset of is not a subset of p Closure property  For any real numbers a and b, a+ b and ab are real numbers. 14. Commutative property  a + b = b + a and ab = ba 15. Associative property  a + (b + c) = (a + b) + c and a(bc) = (ab)c 16. Distributive property  a(b +c) = ab + ac 17. Identity Properties a = a and 1. a = a (zero is the additive identity and 1 is the multiplicative identity)
2 18. Multiplication property of zero  0. a = Additive inverse property  For any real number a, a + a = For every nonzero real number a, a. 1/a = 1. 1/a is the reciprocal (multiplicative inverse) of a. p Properties of opposites  a) 1. a = a b) (a) = a c) (a  b) = b  a 22. relations  symbols that indicate how numbers are related such a <, >, =,, and. 23. Trichotomy property  For any two real numbers a and b, exactly one of the following is true: a< b, a = b, or a > b. p Reflexive property  For any real number a, a = a 25. Symmetric property  For any real numbers a and b, if a = b, then b = a 26. Transitive property  For any real numbers a, b, and c, if a = b and b = c, then a = c. 27. Substitution property  If a = b then a and b may be substituted for one another in any expression involving a or b. 28. Absolute value  the distance of a number a from 0 on the number line. a = { a if a 0 a if a < 0 p Properties of absolute value: a) a o (absolute value is nonnegative) b) a = a (additive inverses have the same absolute value) c) a. b = a. b (the absolute value of a product is the product of the absolute values. d) a/b = a / b, b 0 ( the absolute value of a quotient is the quotient of the absolute values.) 30. Distance between two points  Distance is the absolute value of the their difference or d(a,b) = a  b p Arithmetic expression  writing numbers with operation of math. 32. Order of operations  (note  error on lesson  grouping before powers) Grouping symbols Powers Multiplication and division  left to right Addition and subtraction  left to right
3 p ) Algebraic expression (variable expression)  writing numbers and one or more variables with operations of math. p Value of an algebraic expression  value of arithmetic expression when variable is replaced by real numbers. 35. Domain  set of numbers that are allowed to be used for the variable in an algebraic expression. 36. Simplify  find a simplerlooking equivalent expression. 37. term  single number or the product of a number and one or more variables. 38. factor  the individual numbers or variables in a term. 39. coefficient  the product of the remaining factors of any variable part in the term. 40. like terms  two or more terms that contain the same variables with the same exponents. P2 p. 16 1) exponential expression  a n = a. a. a..... a 2) base  in the expression a n, a is the base or factor. 3) exponent  in the expression a n, n is the exponent or power. p negative integral exponents  If a is a nonzero real number and n is a positive integer, a n = 1 a n p Rules for negative exponents and fractions  If a and b are nonzero numbers and m and n are integers, then (a/b) m = (b/a) m and a m = b n b n a m 6. product rule  a m. a n = a m+n p Zero exponent  If a is nonzero real number, then a 0 = 1 8. Rules for Integral exponents  a) a m a n = a m+n Product rule b) a m = a mn Quotient rule a n c) (a m ) n = a mn Power of a power (continued)
4 d) (ab) n = a n b n Power of a product rule e) (a/b) n = a n / b n Power of a quotient rule p. 22 9) Scientific notation  a number between 1 and 10 times a power of 10. P3 p nth root  if n is a positive integer and a n = b, then a is the nth root of b. If a 2 = b then a is the square root of b If a 3 = b then a is the cube root of b. 2. Exponent 1/n  if n is a positive even integer and a is positive, then a 1/n is the positive real nth root of a and is called the principal nth root of a. if n is positive and odd and a is real then a 1/n is the real nth root of a if n is positive then 0 1/n = 0 p rational exponents  If m and n are positive integers, then a m/n = (a 1/n ) m provided a 1/n is real. p Rules for rational exponents: a and b are real and r and s are rational, powers are real and no denominators are zero. a) a r a s = a r+s b) a r = a rs a s c) (a r ) s = a rs d) (ab) r = a r b r e) (a/b) r = a r b r f) (a/b) r = b r a r g) a r = b s b s a r p radical sign  and the exponent 1/n both indicate the nth root. 6. radical  If n is a positive integer and a is a number for which a 1/n is defined, then the expression = a 1/n. If n = 2, the write a rather than a 7. radicand  the number under the radical sign. 8. index  the n of the radical
5 p rule for converting a m/n  if a is real and m and n are integers for which a m/n = ( ) m = 10. rules for radicals  For any positive integer n and real numbers a and b (b= 0) 1. ab = a b  product rule for radicals 2. a = a  quotient rule for radicals b b p perfect nth power  expression where the nth root of a term is free of radicals. 12. perfect square  square root of a number is free of radicals. 13. perfect cube  cube root of a number is free of radicals. 14. simplified form for radicals of index n a) no perfect nth powers as factors of the radicand b) no fractions inside the radical c) no radicals in a denominator p rationalizing the denominator  removing the radical from the denominator by multiplying by a term that makes the denominator a perfect nth power. p like radicals  radical expression with the same index P4 p polynomial  single term or a finite sum of terms 2. polynomial in x  If n is a nonnegative integer and a 0, a 1, a 2,...a n are real numbers, then a n x n + a n1 x n1 + a n2 x n a 1 x + a 0 is a polynomial in a single variable x. 3. constant  single number 4. monomial  a polynomial with only one term
6 5. binomial  a polynomial with two terms 6. trinomial  a polynomial with three terms p leading coefficient  the coefficient of the first term when polynomial is written in decreasing order from left to right. 8. degree of a polynomial in one variable  highest power of the variable in the polynomial 9. zero polynomial  zero 10) linear polynomial  first degree polynomial 11) quadratic polynomial  seconddegree polynomial 12) cubic polynomial  thirddegree polynomial 13) degree of term with more than one variable  sum of powers of variables 14) degree of polynomial with more than one variable  highest degree of any of its terms. 15) adding and subtracting polynomials  add and subtract like terms p ) multiplying polynomials  multiply each term of the first polynomial by every term of the second polynomial and then combine like terms p ) FOIL  product of two binomials consists of four terms: (a + b) ( c + d) = ac (first terms) + ad (outer terms) + bc (inner terms) + bd (last terms) p ) special products: a) square of a sum  (a + b) 2 = a 2 + 2ab + b 2 b) square of a difference  (a  b) 2 = a 22ab + b 2 c) product of a sum and a difference  (a+b)(ab) = a 2  b 2 p ) conjugates  two expressions with radicals whose product is a rational number p ) division algorithm of polynomials  If the dividend P(x) and the divisor D(x) are polynomials where D(x) is not zero and the degree of P(x) is to the degree of D(x), then there are two polynomials the quotient Q(x) and the remainder R(x), such that P(x) = Q(x) D(x) + R(x) where R(x) = 0 or the degree of R(x) the degree of D(x).
7 p ) value of a polynomial P(x)  number to replace the variable in a polynomial P5 p. 54 1) Factoring out  the process of finding factor that is common to each term and taking it out. 2) Factors  polynomials when multiplied give back the original polynomial 3) Common Factor  a factor that is common to the terms of a polynomial 4) Greatest Common Factor (GCF)  the monomial that includes every number and variable that is a factor of all terms of the polynomial. p. 55 5) Factoring by Grouping  used with a polynomial of 4 terms where the common factors are factored out of the first pair and second pair. p. 56 6) Factoring ax 2 + bx + c with a = 1 Find the two numbers e and f whose product is c and whose sum is b then factor (x + e)(x + f) p. 57 7) Factoring ax 2 + bx + c with a 1 (Split the middle term) a) Find two numbers who sum is b and whose product is ac b) Replace b by the sum of these two numbers c) Factor the resulting fourterm polynomial by grouping p. 58 8) Perfect Square Trinomial  trinomial that results from squaring a sum or a difference. 9) Factoring Special Products: a) Difference of two squares a 2  b 2 = (a + b)(a  b) b) Perfect Square Trinomial a 2 + 2ab + b 2 = (a + b) 2 c) Perfect Square Trinomial a 22ab + b 2 = (a  b) 2 p ) Factoring the Difference and Sum of Two Cubes a) Difference of two cubes a 3  b 3 = (a  b)(a 2 + ab + b 2 ) b) Sum of two cubes a 3 + b 3 = (a + b)(a 2  ab + b 2 ) 11) Factoring by Substitution  when a polynomial involves a complicated expression we a) replace the complicated expression by a single variable b) factor the simpler polynomial c) replace the single variable by the complicated expression.
8 p ) Prime (irreducible over the integers)  polynomials that cannot be factored using integral coefficients. 13) Factoring Completely  writing a polynomial as a product of prime polynomials. P6 p. 65 1) rational expression  ratio of two polynomials in which the denominator is not the zero polynomial 2) domain  set of all real numbers that can be used in place of variable 3) Basic principle of rational numbers: If a, b, and c are integers with b 0 and c 0, then ac = bc a b p. 66 4) reduce to lowest terms  divide out all common factors p. 67 5) multiplying rational numbers  If a/b and c/d are rational numbers, then a. c = ac b d bd p. 68 6) dividing rational numbers  If a/b and c/d are rational numbers with c 0, then a c = a. d b d b c 7) build up a denominator  rename a denominator with a larger value by multiplying the numerator and denominator by the same number to get an equivalent fraction. p. 69 8) least common denominator (LCD)  smallest number that is a multiple of all the denominators. Steps: a) factor each denominator completely b) write a product using each factor that appears in a denominator c) for each factor, use the highest power of that factor that occurs in the denominator 9) adding and subtracting rational numbers  If a/b and c/d are rational numbers, then a + c = a + c and a  c = a  c b b b b b b
9 p ) complex fraction  a fraction having rational expression in the numerator, denominator, or both 11) simplifying a complex fraction  multiply the numerator and denominator by the LCD of all of the denominators. 12) simplifying complex fractions with negative exponents  a) change all terms to positive powers b) multiply numerator and denominator by LCD of the denominators
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