Introduction to Polynomials - Mathematics, Class 10 Video Lecture

- 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.