- WELCOME TO A VIDEO ON EVALUATING
POLYNOMIAL EXPRESSIONS AND POLYNOMIAL FUNCTIONS.
YOU CAN SEE OUR GOALS LISTED BELOW.
LET'S GO AHEAD AND GET STARTED.
AN EXPRESSION IN WHICH ALL OF THE TERMS
ARE MONOMIALS CONNECTED WITH ADDITION OR SUBTRACTION
IS CALLED A POLYNOMIAL EXPRESSION.
WE WERE WORKING WITH POLYNOMIAL EXPRESSIONS
WHEN WE WERE COMBINING LIKE TERMS IN THE PREVIOUS VIDEO.
IF WE ARE GIVEN THE VALUES OF THE VARIABLES
IN A POLYNOMIAL EXPRESSION,
WE CAN EVALUATE THE POLYNOMIAL EXPRESSION.
SO FOR EXAMPLE, IF WE HAVE THE EXPRESSION 5X -12 AND X = 17,
WE CAN REPLACE X = 17, WHICH IS CALLED SUBSTITUTION,
AND THEN EVALUATE THE EXPRESSION.
SO INSTEAD OF 5X -12 WE HAVE 5 X 17 MINUS 12 WHICH EQUALS 73.
LET'S TRY A COUPLE OF OUR OWN.
WE WANT TO EVALUATE THESE EXPRESSIONS
GIVEN X = -3, Y = 7, AND Z = -2.
SO WE WOULD HAVE, INSTEAD OF -5 x X, WE'D HAVE -5 X -3 +.
INSTEAD OF 3 X Y WE'D HAVE 3 X 7 - 15.
NOW WE SIMPLIFY THIS EXPRESSION TO GET 15 + 21 - 15
WHICH IS EQUAL TO 21.
NOW WE CAN'T EASILY CHECK THESE
WITH THE USE OF THE GRAPHING CALCULATOR.
LET ME SHOW YOU THAT.
THE FIRST THING WE HAVE TO DO
IS STORE THE VALUES INTO THE VARIABLES.
SO IN ORDER TO STORE -3 IN FOR X, WE PRESS -3, STORE, X, ENTER.
Y IS EQUAL TO 7, SO I PRESS 7 STORE.
NOW TO ACCESS THE Y WE HIT ALPHA 1 ENTER, AND Z IS EQUAL TO -2,
SO WE PRESS -2 STORE ALPHA 2 WILL GIVE US Z.
PRESS ENTER.
NOW WE CAN JUST TYPE IN THE GIVEN EXPRESSION
-5X + 3Y - 15 TO VERIFY OUR ANSWER.
OKAY, LET'S TRY ANOTHER.
AGAIN, THE FIRST STEP IS TO REPLACE THE VARIABLES
WITH THE GIVEN VALUES,
SO HERE WE HAVE, INSTEAD OF 2 X X SQUARED
WE HAVE 2 X -3 SQUARED PLUS 5 X Y, BUT Y IS 7 - Z,
BUT SINCE Z IS EQUAL TO -2.
LET'S SIMPLIFY OUR EXPONENTS FIRST.
-3 SQUARED WOULD EQUAL 9.
-2 CUBED WOULD EQUAL -8.
LET'S CONTINUE TO SIMPLIFY.
HERE WE'D HAVE 18 + 35 + 8 WHICH IS EQUAL TO 61.
AGAIN, LET'S VERIFY THIS WITH THE GRAPHING CALCULATOR.
SINCE WE ALREADY HAVE THE VALUES STORED FOR THE VARIABLES,
WE CAN JUST ENTER IN THE EXPRESSION
2X SQUARED + 5Y - Z TO THE 3RD POWER,
AND THAT CHECKS OUR WORK.
OKAY, NOW LET'S TALK ABOUT POLYNOMIAL FUNCTIONS.
A FUNCTION IN WHICH ALL OF THE TERMS ARE MONOMIALS
CONNECTED WITH ADDITION OR SUBTRACTION
IS CALLED A POLYNOMIAL FUNCTION.
A LINEAR FUNCTION IS DESCRIBED BY A POLYNOMIAL OF DEGREE 1,
A QUADRATIC FUNCTION IS DESCRIBED
BY A POLYNOMIAL OF DEGREE 2, A CUBIC, DEGREE 3,
AND A QUARTIC DEGREE 4.
SO HERE WE HAVE A POLYNOMIAL FUNCTION OF DEGREE 3,
AND WE'RE ASKED TO FIND THE FUNCTION VALUES.
SO IT'S ESSENTIALLY THE SAME AS EVALUATING AN EXPRESSION
EXCEPT THE NOTATION IS DIFFERENT.
HERE WE HAVE A FUNCTION
WHERE BEFORE WE JUST HAD THE EXPRESSION.
HERE WE CAN SEE THE X HAS BEEN REPLACED WITH -5,
AND THIS IS EXACTLY HOW WE EVALUATE THIS FUNCTION
WHEN X IS EQUAL TO -5, BY PERFORMING SUBSTITUTION.
SO WE HAVE -5 CUBED + 3 x -5 SQUARED - 9 TIMES -5 - 13.
SO NOW WE NEED TO SIMPLIFY THIS.
-5 CUBED WOULD EQUAL -125.
-5 SQUARED WOULD EQUAL 25 X 3.
IT WOULD BE 75.
-9 X -5 WOULD BE 45 AND THEN -13.
WE SIMPLIFY ALL OF THIS, WE SHOULD OBTAIN -18.
SO IN CONCLUSION, P OF -5 IS EQUAL TO -18.
AND I WANT TO MAKE THE CONNECTION
THAT THIS ACTUALLY REPRESENTS A POINT ON THIS FUNCTION,
AND THAT POINT WOULD BE WHEN X IS EQUAL TO -5,
Y WOULD EQUAL -18.
SO IF WE GRAPH THIS FUNCTION,
THIS POINT WOULD BE ON THE GRAPH.
NOW AGAIN, LET'S VERIFY THIS ON THE GRAPHING CALCULATOR.
NOW WE COULD DO IT THE SAME WAY WE DID THE EXPRESSION,
BUT SINCE IT IS A FUNCTION IN ONE VARIABLE,
LET'S GO AHEAD AND TYPE IT INTO Y1,
AND LET'S UTILIZE THE TABLE FUNCTION THIS TIME.
SO IF WE PRESS 2ND GRAPH,
MY TABLE IS SET ON THE AUTOMATIC FEATURE,
SO I CAN JUST SCROLL UP TO -5
AND SEE THAT -18 IS THE VALUE OF THE FUNCTION WHEN X IS -5,
SO THAT VERIFIES OUR ANSWER.
AND IN FACT, IF WE WANTED TO TAKE THE SHORTCUT FOR NUMBER 2,
WE COULD SCROLL DOWN TO X = 3
AND SEE THAT WHEN X IS EQUAL TO 3 Y IS EQUAL TO 14.
THEREFORE, P OF 3 IS EQUAL TO 14.
NOW, I WOULD ENCOURAGE YOU TO VERIFY THIS BY HAND
AS WE DID ON NUMBER ONE,
BUT TO SAVE TIME, I'M GOING TO GO AHEAD
AND GO ON TO THE NEXT PROBLEM.
HERE'S AN APPLICATION OF A DEGREE 2 POLYNOMIAL FUNCTION,
BETTER KNOWN AS A QUADRATIC FUNCTION.
IT STATES THAT DISTANCE S OF T IN FEET
TRAVELED BY A FREE-FALLING BODY FROM REST IN T SECONDS
IS APPROXIMATELY S OF T EQUALS 16 T SQUARED.
IF A SKYDIVER JUMPS OUT OF A PLANE
AND FREE FALLS FOR A TOTAL OF 8 SECONDS
BEFORE PULLING THE PARACHUTE,
WHAT DISTANCE HAS THE SKYDIVER TRAVELED
BEFORE PULLING THE PARACHUTE?
WE'RE GIVEN THE TIME OF 8 SECONDS,
SO WE NEED TO FIND S OF 8 WHICH IS EQUAL TO 16 X T SQUARED,
BUT NOW WE KNOW T IS 8,
SO THIS WOULD EQUAL 16 X 64 WHICH IS EQUAL TO 1,024,
AND THE UNITS FOR S OF T IS IN FEET.
SO THE SKYDIVER HAS FALLEN 1,024 FEET.
I HOPE YOU FOUND THIS LESSON USEFUL.
THANK YOU FOR WATCHING AND HAVE A NICE DAY.