Lesson 3 Algebraic expression: - the result obtained by applying operations (+, -,, ) to a collection of numbers and/or variables o

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1 Lesson 3 Algebraic expression: - the result obtained by applying operations (+, -,, ) to a collection of numbers and/or variables o o ( 1)(9) 3 ( 1) Evaluate the second expression at the left, if x = -1 and y = 9 - in order to find the value of an algebraic expression, specific numbers must be substituted in for the variables - algebraic expressions can only be simplified, NOT solved Basic Monomial (of one variable): - an expression of the form In general, a monomial may contain more than one variable. For example: where the coefficient is a real number and the exponent nonnegative integer is a Since only nonnegative integers are used for exponents, the following are not monomials. c c n,, c x n n x x Binomial (of one variable): - sum/difference of two monomials o Trinomial: - Sum/difference of three monomials o 1

2 Polynomial: - a sum of any number of monomials - each monomial is called a term - the exponent of the variable in each term is called the degree of that term. - if a term has more than one variable, the sum of all the variable exponents is the degree of the term - the degree of the polynomial is largest degree from all the terms - the leading coefficient of the polynomial is the coefficient of the term with the largest degree (a) Find the degree of each term, the degree of the polynomial, and the leading term for this polynomial: (b) Find the degree of this polynomial: Adding polynomials: - combine like terms (terms with the same variables to the same powers) Example 1: Express as a polynomial; simplify your answer completely. - Write answer in descending order of powers. Distributive Property (monomial times polynomial): - and (The second half of the previous statement is known as factoring.) Multiply: - ***After the factor of has been distributed, the expression goes from a product of two factors to a sum of two terms.*** - The distributive property is a method to convert between a product (multiplication) and a sum (addition).

3 Subtracting polynomials: - use the distributive property, then combine like terms - here the distributive property is used to multiply by each term in a subtracted polynomial Importance of using parentheses: - if you are subtracting one term, parentheses are not required, but can still be used - when you are subtracting a quantity, parentheses MUST BE USED at the start, because you must subtract every term in the quantity Example : Express as a polynomial; simplify your answer completely. a. b. Multiplying a monomial and a polynomial: - use the distributive property and the properties of exponents - the distributive property can be used as above to distribute one term or it can be used to distribute multiple terms - remember that order and grouping are immaterial when multiplying 3

4 Example 3: Multiply (a) (b) Distributive Property (polynomial times polynomial): - - after using the distributive property, the expression goes from a product of two factors to a sum of four terms. Use the distributive property to multiply: (x 3)(4x 5) - Shortcut for multiplying binomials: FOIL - A variation of FOIL is used for multiplying two polynomials where one or more has more than terms Example 3: Express as a polynomial (simplify your answer completely combining like terms) a. b. b. 4

5 c. d. d. e. f. Dividing a polynomial by a monomial: - write each term in the numerator as a separate fraction over the denominator - simplify each of the resulting fractions by cancelling common factors or by using the Quotient Rule for Exponents (Examples on next page.) 5

6 Example 4: Express as a polynomial (simplify your answer completely) a. b. Special Products of binomials: 1. ***( a b)( a b) a b Product of a sum and a difference or (1 st) ( nd). ***( a b) a ab b Squaring a binomial or (1 ) (1 )( ) ( st st nd nd) *** Memorize the patterns, but understand why these short-cuts work.*** Example 5: Express as a polynomial (simplify your answer completely) a. b. b. 6

7 c. d. (Cannot use exponent properties.) d. e. ( ) ( ) f. 7