Slide 1:ALGEBRA Engr. Lizette Ivy G. Catadman Instructor / Lecturer LECTURE NOTES & PROBLEM SETS: PART 2


Slide 2:TYPE PRODUCTS SUM AND DIFFERENCE OF TWO TERMS (BINOMIALS) SQUARE OF A BINOMIAL


Slide 3:TYPE PRODUCTS PRODUCT OF TWO BINOMIALS CASE 1 PRODUCT OF TWO BINOMIALS CASE 2


Slide 4:TYPE PRODUCTS CUBE OF A BINOMIAL SQUARE OF A TRINOMIAL


Slide 5:TYPE PRODUCTS SQUARE OF A MULTINOMIAL PRODUCT OF A BINOMIAL AND A TRINOMIAL


Slide 6:TYPE PRODUCTS BINOMIAL RAISED TO POWER n


Slide 7:TYPE PRODUCTS BINOMIAL RAISED TO POWER n PASCAL’S TRIANGLE


Slide 8:FACTOR TYPES COMMON FACTOR DIFFERENCE OF TWO SQUARES


Slide 9:FACTOR TYPES PERFECT TRINOMIAL SQUARE SUM AND DIFFERENCE OF TWO CUBES


Slide 10:FACTOR TYPES TRINOMIALS WITH DISTINCT FACTORS THE TRINOMIAL MUST BE ARRANGED IN DESCENDING POWERS OF THE GIVEN VARIABLE. IN THE ABOVE EQUATIONS, IT IS THE VARIABLE x.


Slide 11:FACTOR TYPES TRINOMIALS WITH DISTINCT FACTORS IF THE LAST SIGN IN THE TRINOMIAL IS (+), THE SIGNS IN THE BINOMIAL FACTORS ARE THE SAME. IF THE MIDDLE TERM OF THE TRINOMIAL IS (+), THE SIGNS IN BOTH BINOMIALS ARE (+). IF THE MIDDLE TERM OF THE TRINOMIAL IS (-), THE SIGNS IN BOTH BINOMIALS ARE (-).


Slide 12:FACTOR TYPES TRINOMIALS WITH DISTINCT FACTORS IF THE LAST SIGN IN THE TRINOMIAL IS (-), THE SIGNS IN THE BINOMIALS ARE DIFFERENT. TAKE THE PRODUCT OF THE INNER TERMS AND OUTER TERMS OF THE FACTORS (BINOMIALS). WHICHEVER HAVE THE HIGHER RESULT WILL CARRY THE SIGN OF THE MIDDLE TERM OF THE TRINOMIAL.


Slide 13:MORE COMPLICATED FACTOR TYPES FACTORING BY REARRANGEMENT AND GROUPING CASE 1: IF A POLYNOMIAL HAS FOUR TERMS, IT MAY BE FACTORED OUT BY FIRST, SEPARATING IT INTO TWO GROUPS AND FACTORING EACH GROUP SEPARATELY. IF EACH OF THE TWO GROUPS HAS A COMMON FACTOR, THEN THE POLYNOMIAL CAN BE FACTORED OUT.


Slide 14:MORE COMPLICATED FACTOR TYPES FACTORING BY REARRANGEMENT AND GROUPING CASE 2: IF A POLYNOMIAL HAS FOUR TERMS, SEPARATE THREE TERMS THAT WILL FORM A TRINOMIAL PERFECT SQUARE. IF THE EVENTUAL RESULT IS IN THE FORM OF THE DIFFERENCE OF TWO SQUARES, THE POLYNOMIAL CAN BE FACTORED OUT.


Slide 15:MORE COMPLICATED FACTOR TYPES FACTORING BY REARRANGEMENT AND GROUPING CASE 1: CASE 2:


Slide 16:MORE COMPLICATED FACTOR TYPES SUM AND DIFFERENCE OF TWO ODD POWERS DIFFERENCE OF TWO EVEN POWERS


Slide 17:MORE COMPLICATED FACTOR TYPES DIFFERENCE OF TWO EVEN POWERS


Slide 18:MORE COMPLICATED FACTOR TYPES FACTORING BY ADDING OR SUBTRACTING A CERTAIN TERM TO FORM A TRINOMIAL PERFECT SQUARE ADDING AND SUBTRACTING THE SAME TERM IS EQUIVALENT TO ADDING NOTHING, BUT THIS MAKES THE FORM FACTORABLE AS A DIFFERENCE OF TWO SQUARES.


Slide 19:MORE COMPLICATED FACTOR TYPES FACTORING BY ADDING OR SUBTRACTING A CERTAIN TERM TO FORM A TRINOMIAL PERFECT SQUARE


Slide 20:MORE COMPLICATED FACTOR TYPES FACTORING BY SYNTHETIC DIVISION OR TRIAL DIVISION WRITE DOWN ALL THE NUMERICAL COEFFICIENTS OF THE POLYNOMIAL, INCLUDING THE ZERO COEFFICIENTS. BY TRIAL AND ERROR, FIND A NUMBER WHICH WILL EQUATE THE LAST COLUMN TO ZERO.


Slide 21:MORE COMPLICATED FACTOR TYPES FACTORING BY SYNTHETIC DIVISION OR TRIAL DIVISION COPY THE REMAINING NUMBERS EXCEPT THE ZERO ON THE LAST COLUMN. RECONSTRUCT THE POLYNOMIAL. THE DEGREE OF THE POLYNOMIAL MUST DECREASE BY ONE.


Slide 22:MORE COMPLICATED FACTOR TYPES FACTORING BY SYNTHETIC DIVISION OR TRIAL DIVISION ROOT FACTOR IS: A+1


Slide 23:HIGHEST COMMON FACTOR OR HCF EXPRESSION OF HIGHEST DEGREE WHICH IS A FACTOR OF EACH GIVEN EXPRESSION. LOWEST COMMON MULTIPLE OR LCM EXPRESSION OF LOWEST DEGREE WHICH IS EXACTLY DIVISIBLE BY ALL OF THEM.


Slide 24:HCF AND LCM