VIEWS: 18 PAGES: 4 POSTED ON: 5/3/2011 Public Domain
POLYNOMIALS I. Exponents A. Definitions Exponents are shorthand notation for repeated multiplication a a n a a a . n factors Zero exponent b0 1 1 Negative exponent b n bn To see why the zero exponent and negative exponents are defined this way consider the pattern in the example below. In the left column the exponents decrease by 1 and in the right column the numbers decrease by a factor of 10. 103 1000 10 2 100 101 10 10 0 1 1 1 10 1 1 10 10 1 1 10 2 2 100 10 1 1 10 3 3 1000 10 B. Rules of Exponents Product rule (same base) a ma n a mn am m n n a Quotient rule (same base) a Product rule (same exponent) ( ab) m a m b m m a am Quotient rule (same exponent) m b b Power rule a m n a mn II. Polynomials A polynomial has terms consisting of constants and variables raised to positive integer powers. Polynomials are classified according to the number of terms they have. Monomials have one term, binomials have two terms and trinomials have three terms. The degree of a polynomial in one variable is the greatest exponent. The degree of a polynomial of several variables is the greatest sum of the exponents that occurs in any of the terms. III. Addition/Subtraction Like terms have the same variables with the same exponents. Polynomials are added/subtracted by combining like terms. IV. Multiplication A. Multiply a polynomial by a monomial Multiply each term of the polynomial by the monomial. This is the same as saying use the distributive rule. B. Multiply a polynomial by a polynomial FOIL Method. When multiplying binomials use first, outer, inner and last and combine like terms. General Method. Multiply each term of the second polynomial by each term of the first polynomial and combine like terms. V. Factoring Choose a factoring technique based on the number of terms. A. Greatest common factor Always factor out the GCF before moving on to other techniques. Factor out the largest number that divides into each term evenly. Factor out the variables that are common to all the terms using the lowest exponent. B. Factoring formulas (two terms) Difference of squares x 2 a 2 ( x a )( x a ) Difference of cubes x 3 a 3 ( x a )( x 2 ax a 2 ) Sum of cubes x 3 a 3 ( x a )( x 2 ax a 2 ) C. Factoring trinomials (three terms) When the polynomial has three terms use the trial and error method. Leading coefficient is a 1. Since x 2 (a b) x ab ( x a)( x b) to factor x 2 bx c we must find numbers whose sum is b and whose product is c. In other words find two numbers that multiply to the last term and add to give the coefficient of the middle term. Leading coefficient is not a 1. Firsts must multiply to give the first term. Lasts must multiply to give the third term. The inner and outer must sum to give the second term. Comments: (1) The AC test states that ax 2 bx c is factorable only if there are two numbers whose product is ac and whose sum is b. For example 3x 2 2 x 5 is not factorable since there are no numbers, which multiply to 15 and add to 2. (2) The slide technique as shown in the example below has less trial and error for the case in which the leading coefficient is not 1. 6 x 2 x 12 Multiply the last term by the x 2 x 72 leading coefficient. Factor and then divide both 9 8 numbers by the leading x x 6 6 coefficient. 3 4 Reduce the fractions. x x Multiple the denominators up 2 3 (2 x 3)(3 x 4) to the variables. D. Factoring by grouping (four or more terms) Rearrange the terms (if necessary) so that the first two terms have a common factor and the last two terms have a common factor. Factor out the GCF from the first two terms and the second two terms. Factor a binomial factor from the remaining terms. VI. Solving Equations Use the zero factor property. If ab 0 then a 0 or b 0 . VII. Division A. Division by a monomial Divide each term by the monomial B. Division by a polynomial Use the long division algorithm. C. Synthetic division Used to divide a polynomial by a binomial of the form x c . Ex. Divide 2x 4 3x 2 5x 7 by x 3 . 3 2 0 3 5 7 6 18 45 120 2 6 15 40 113 113 2 x 3 6 x 2 15x 40 x3 College Algebra Home Page