- WELCOME TO AN INTRODUCTION TO POLYNOMIALS.
THE GOAL OF THIS VIDEO IS TO IDENTIFY POLYNOMIALS
AND KNOW THE DIFFERENT VOCABULARY TERMS USED
WHEN DISCUSSING POLYNOMIALS.
SO, LET'S START OFF WITH SOME VOCABULARY.
A TERM CAN BE A NUMBER, A VARIABLE, A PRODUCT OF NUMBERS
AND/OR VARIABLES, OR A QUOTIENT OF NUMBERS AND/OR VARIABLES.
HERE ARE SOME EXAMPLES OF SOME TERMS,
5X, NEGATIVE 3A SQUARED, B CUBED, 1/5,
NEGATIVE 8 OVER X SQUARED, M OVER N,
7X TO THE POWER OF NEGATIVE 3.
THESE ARE ALL TERMS BECAUSE THEY SATISFY THESE CONDITIONS.
NOW, MONOMIAL IS A REAL NUMBER AND/OR THE PRODUCT OF VARIABLES
WITH NON-NEGATIVE EXPONENTS.
SO, SOME OF THESE TERMS ARE MONOMIALS
AND SOME OF THEM AREN'T, 5X, NEGATIVE 3A SQUARED, B CUBED
AND 1/5 ARE THE ONLY TERMS THAT ARE ALSO MONOMIALS.
THESE THREE TERMS ARE NOT MONOMIALS
BECAUSE THESE TWO INVOLVE QUOTIENTS OF VARIABLES
AND THIS ONE HAS A NEGATIVE EXPONENT.
A POLYNOMIAL IS A MONOMIAL OR A SUM OF MONOMIALS.
REMEMBER THAT SUBTRACTION CAN BE REWRITTEN AS ADDITION,
SO IT MAY SHOW UP AS A SUBTRACTION OR MONOMIALS.
SO, FOR EXAMPLE, 5X + 3 IS A POLYNOMIAL.
-X CUBED + 5Y SQUARED - 8 IS ALSO A POLYNOMIAL.
NOW, POLYNOMIALS DO HAVE FURTHER CLASSIFICATION
BASED UPON THE NUMBER OF TERMS IN THE EXPRESSION.
SO FOR EXAMPLE, IF WE HAVE MONOMIALS,
MONO MEANS 1 SO WE HAVE THE 1 TERM.
ALL OF THESE ARE MONOMIALS.
BINOMIALS CONSIST OF TWO MONOMIALS
CONNECTED WITH ADDITION OR SUBTRACTION.
HERE ARE SOME EXAMPLES OF BINOMIALS.
IF THE POLYNOMIAL HAS 3 TERMS, IT'S CALLED A TRINOMIAL.
HERE ARE TWO EXAMPLE OF TRINOMIALS
AND ONCE A POLYNOMIAL GETS PAST THREE TERMS,
WE USUALLY JUST CALL IT A POLYNOMIAL.
THE WORD POLY MEANS MANY.
LET'S GO AHEAD AND CONSIDER THIS POLYNOMIAL.
THE FIRST THING WE SHOULD BE ABLE TO DO
IS LIST THE TERMS OF THE POLYNOMIAL
AND THE ONLY THING YOU HAVE TO BE CAREFUL ABOUT
IS DEALING WITH THE SUBTRACTION.
SO, THE FIRST TERM WOULD BE 4X SQUARED, Y CUBED.
THE SECOND TERM--REMEMBER THE ORIGINAL DEFINITION
OF A POLYNOMIAL WAS THE SUM OF MONOMIALS.
SO, THIS IS THE SAME AS PLUS A NEGATIVE 12X CUBED.
SO, THE TERM IS NEGATIVE 12X CUBED.
I JUST THINK THAT YOU TAKE THE SIGN WITH IT.
IF IT'S A MINUS SIGN, SHOOT IT AS A NEGATIVE.
THE THIRD TERM WOULD BE 24XY.
FOURTH TERM WOULD BE X AND THE FIFTH TERM AGAIN
WOULD BE A NEGATIVE 15.
NEXT, WE WANT TO TALK ABOUT DEGREE.
THE DEGREE OF A TERM IS THE NUMBER OF VARIABLE FACTORS
IN THAT TERM.
YOU CAN ALSO SAY THE DEGREE OF A TERM
IS THE SUM OF THE EXPONENTS ON THE VARIABLES.
SO, THE DEGREE OF THIS FIRST TERM,
BECAUSE THERE'S TWO FACTORS OF X HERE AND THREE FACTORS OF Y,
THE DEGREE OF THIS TERM WOULD BE 5.
YOU CAN SEE AN EASY WAY TO DO THAT
IS JUST ADD THE POWERS ON THE VARIABLES.
THIS TERM HAS DEGREE 3.
THIS TERM HAS ONE FACTOR OF X, ONE FACTOR OF Y,
THEREFORE, THE TERM HAS DEGREE TWO.
THIS WOULD BE X TO THE FIRST, SO IT'S A DEGREE ONE TERM.
THIS DOES NOT HAVE ANY FACTORS OF A VARIABLE,
THEREFORE THE DEGREE IS ZERO.
OR YOU COULD ALSO THINK OF IF YOU HAD TO INCLUDE A VARIABLE,
IT WOULD HAVE TO BE SOMETHING LIKE X TO THE ZERO POWER
BECAUSE THAT WOULD BE EQUAL TO ONE,
WHICH WOULD NOT CHANGE THE TERM.
THE LEADING TERM OF A POLYNOMIAL
IS THE TERM OF THE HIGHEST DEGREE.
SO, THIS TERM IN THE POLYNOMIAL HAS DEGREE 5,
THEREFORE WE CALL IT THE LEADING TERM.
NOTICE THE TERMS ARE IN ORDER
FROM HIGHEST DEGREE TO LOWEST DEGREE
AND THIS IS CALLED DESCENDING ORDER.
CONSIDERING THE SAME POLYNOMIAL,
THE DEGREE OF A POLYNOMIAL
IS THE SAME AS THE DEGREE OF THE LEADING TERM.
REMEMBER, THIS IS THE LEADING TERM.
IT HAD DEGREE 5, THEREFORE THE DEGREE OF THE ENTIRE POLYNOMIAL
IS DEGREE 5.
NEXT, THE COEFFICIENT OF A TERM
IS THE CONSTANT FACTOR OF THE NUMBER PART OF THE TERM.
SO, THE COEFFICIENT OF THIS LEADING TERM WOULD BE 4
THE COEFFICIENT OF THE SECOND TERM WOULD BE -12.
THE COEFFICIENT WOULD BE 24 FOR THIS TERM.
THIS COEFFICIENT WOULD BE 1
AND THIS COEFFICIENT WOULD BE NEGATIVE 15.
THE LEADING COEFFICIENT IS THE COEFFICIENT OF THE LEADING TERM
SO IN THIS CASE--
THEREFORE IN THIS CASE, THE LEADING COEFFICIENT IS 4.
OKAY, LET'S SEE IF WE CAN PUT ALL THESE PIECES TOGETHER.
WE'RE GIVEN A POLYNOMIAL.
WE WANT TO FIRST LIST THE TERMS,
2A CUBED, B SQUARED IS THAT FIRST TERM,
7A SQUARED, B SQUARED IS OUR SECOND TERM.
OUR THIRD TERM IS NEGATIVE 0.5AB SQUARED.
THE FOURTH TERM IS 25B AND THE FIFTH TERM IS A -21.
REMEMBER THE COEFFICIENTS ARE THE CONSTANT FACTORS
OF THE TERM OR THE NUMBER PART.
SO, THIS COEFFICIENT IS 2.
THIS COEFFICIENT IS 7.
THIS COEFFICIENT IS NEGATIVE 0.5, 25, AND NEGATIVE 21.
NEXT, THE DEGREE OF EACH TERM
IS THE NUMBER OF VARIABLE FACTORS
OR THE SUM OF THE EXPONENTS ON THE VARIABLES.
SO, THIS TERM HAS DEGREE 5.
THIS TERM HAS DEGREE 4.
THIS TERM HAS DEGREE 3,
TWO FACTORS OF B AND ONE FACTOR OF A.
THIS HAS ONE FACTOR OF B.
THIS HAS ZERO VARIABLE FACTORS, SO IT'S DEGREE ZERO.
THE LEADING TERM IN THIS CASE,
BECAUSE IT IS IN DESCENDING ORDER, WOULD BE THE FIRST TERM.
THAT'S NOT ALWAYS THE CASE,
SO YOU WANT TO MAKE SURE YOU CHECK TO CHOOSE THE TERM
WITH THE HIGHEST DEGREE.
THE LEADING COEFFICIENT THEN OF COURSE IS 2
AND THE DEGREE OF THE POLYNOMIAL
IS THE SAME AS THE DEGREE OF THE LEADING TERM, WHICH IS 5.
HOPEFULLY, ALL OF THIS NEW VOCABULARY MAKES SENSE
AND YOU FOUND THIS VIDEO HELPFUL.
THANK YOU AND HAVE A GOOD DAY.