1.3 Algebraic Expressions Terminology. Notation. Polynomials Addition & Subtraction Multiplication & Division

Terminology  Set  Collection of objects (elements)  Usually denoted by capital letters (R, S, T…)  Elements are typically denoted with lower case letters (a, b, c, d, …)  R typically denotes the set of real numbers  Z typically denotes the set of integers 2

Notation Notation or TerminologyMeaningExamples a is an element of S a is not an element of S S is a subset of Tevery element of S is an element of T Z is a subset of R Constanta letter or symbol that represents a specific element of a set Variablea letter or symbol that represents any element of a set Let x denote any real number Equal =two sets or elements of a set are identical a=b, S=T Not Equaltwo sets or elements of a set are not identical 3

Polynomials  Definition  a polynomial in x is a sum of the form: a n x n + a n-1 x n-1 + … + a 1 x + a 0  a monomial is an expression of the form ax n, where a is a real number and n is a non-negative integer  A binomial is a sum of two monomials  A trinomial is the sum of three monomials  The highest value for n determines the degree of the polynomial  The coefficient, a, associated with the highest value of n is the leading coefficient 4

Polynomials (cont.) ExampleLeading CoefficientDegree 3x 4 + 5x 3 + (-7)x x 8 + 9x 2 + (-2)x18 -5x x

Polynomials (cont.)  Adding  Subtracting 6

Polynomials (cont.)  Multiplying Polynomials 7

Polynomials (cont.) Special Product FormulasExample (x + y)(x - y) = x 2 – y 2 (2a + 3)(2a – 3) = (2a) 2 – 3 2 = 4a (x ±y) 2 = x 2 ± 2xy + y 2 (2a – 3) 2 = (2a) 2 – 2(2a)(3)+3 2 = 4a a + 9 (x ± y) 3 = x 3 ± 3x 2 y + 3xy 2 ± y 3 (2a + 3) 3 = (2a) 3 + 3(2a) 2 (3) + 3(2a)(3) 2 +(3) 3 =8a a a

Polynomials (cont.)  Dividing a Polynomial by a Binomial 9

Practice Problems  Page 43 Problems