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- WELCOME TO A LESSON ON HOW TO USE THE 0 FEATURE
OF THE TI84 GRAPHING CALCULATOR TO FIND THE REAL RATIONAL ZEROS
OR ROOTS OF A POLYNOMIAL FUNCTION.
THIS FIRST EXAMPLE WILL BE PRETTY STRAIGHTFORWARD,
AND THEN THE 2nd EXAMPLE-- AND THEN IN THE 2nd VIDEO
WE'LL TAKE A LOOK AT ONE THAT'S A LITTLE BIT MORE CHALLENGING.
NORMALLY, WHEN ASKED TO FIND THE ZEROS OF A POLYNOMIAL FUNCTION,
WE START BY LISTING ALL THE POSSIBLE RATIONAL ZEROS
OF THE FUNCTION.
THE LIST OF POSSIBLE RATIONAL ZEROS
COME FROM THE RATIO OF THE FACTORS OF THE CONSTANT TERM,
IN THIS CASE, 15,
TO THE FACTORS OF THE LEADING COEFFICIENT,
WHICH IN THIS CASE, WOULD JUST BE 1.
SO BEFORE WE TAKE A LOOK AT THE CALCULATOR,
LET'S MAKE A LIST OF ALL THE POSSIBLE RATIONAL ZEROS,
AND THEN WE'LL FIND THEM GRAPHICALLY.
SO THE FACTORS OF 15 WOULD BE +/-1, +/-3, +/-5, AND +/-15.
ALL OF THESE WOULD DIVIDE EVENLY INTO 15.
AND THEN FOR THE FACTORS OF 1 WE WOULD JUST HAVE +/-1.
NOW, IF WE FORM A RATIO OR A FRACTION USING THESE FACTORS,
WE CAN SEE THAT WITH A DENOMINATOR OF +/-1,
THE POSSIBLE RATIONAL ZEROS WOULD JUST BE THE FACTORS OF 15.
SO AGAIN, WE'D HAVE JUST +/-1, +/-3, +/-5, AND +/-15.
SO THESE ARE THE POSSIBLE RATIONAL ZEROS
OR THE VALUES OF X THAT WOULD MAKE THIS FUNCTION = 0.
WHEN WE TALK ABOUT FUNCTION VALUES,
WE'RE TALKING ABOUT Y VALUES.
SO WE'RE LOOKING FOR THE X VALUES WHERE Y WOULD BE = 0.
GRAPHICALLY, THOSE WOULD BE THE X INTERCEPTS.
SO IF WE GRAPH THIS FUNCTION,
WE CAN FIND THE REAL ZEROS AS X INTERCEPTS,
BUT WE CAN ONLY FIND THE EXACT VALUES GRAPHICALLY
OF THE REAL AND RATIONAL ZEROS.
SO LET'S TAKE A LOOK AT THIS GRAPH.
SO FROM THE HOME SCREEN WE'RE GOING TO PRESS
THE Y = BUTTON HERE,
AND THEN WE'RE GOING TO TYPE IN THE FUNCTION.
NOTICE, Y1 = IS ALREADY THERE.
SO WE'RE JUST GOING TO TYPE IN THE RIGHT SIDE HERE, X CUBED.
HERE'S THE X KEY.
HERE'S THE EXPONENT KEY.
SO THERE IS X CUBED - 3X.
WE COULD PRESS EXPONENT 2
OR THIS BUTTON HERE IS THE SQUARED KEY - 13X + 15.
NOW, TO MAKE SURE WE HAVE THIS INNER WINDOW
ALONG THE X AND Y AXIS FROM -10 TO 10,
LET'S GO AHEAD AND PRESS ZOOM 6.
NOTICE HOW THERE ARE THREE X INTERCEPTS.
SO WE KNOW THERE'S THREE REAL ZEROS,
BUT NOW, WE'LL USE THE 0 FEATURE TO SEE IF THEY'RE ALSO RATIONAL.
TO DO THIS, WE'RE GOING TO PRESS 2nd TRACE
AND THEN OPTION 2 IS THE 0 FEATURE.
SO PRESS THE NUMBER 2.
WE CAN ONLY FIND ONE OF THESE AT A TIME,
AND FOR EACH ONE THEY'RE GOING TO ASK US TO FIND
THE LEFT BOUND AND THE RIGHT BOUND.
SO IF WE WANT TO FIND THIS X INTERCEPT HERE TO THE LEFT,
MEANS WE WANT TO PLACE THE CURSOR BELOW THIS X INTERCEPT.
SO WE'RE GOING TO PRESS THE LEFT ARROW
UNTIL WE SEE THE CURSOR BELOW THAT X INTERCEPT.
SO RIGHT IN HERE PRESS ENTER. RIGHT BOUND IN THIS CASE
IS GOING TO BE ABOVE THE X INTERCEPT AGAIN,
BECAUSE ON THIS INTERVAL THE FUNCTION IS INCREASING.
WE'RE GOING UPHILL.
SO THIS POINT HERE IS TO THE RIGHT OF THE X INTERCEPT.
PRESS ENTER.
WHEN IT SAYS, "GUESS," WE CAN JUST PRESS ENTER AGAIN,
AND NOTICE HOW THE X INTERCEPT IS THE POINT -3, 0.
THEREFORE, WE HAVE THE 0 OF X = -3.
LET'S WRITE THAT DOWN.
NOW, WE'LL GO AHEAD AND FIND THE OTHER TWO.
SO WE'RE GOING TO PRESS 2nd TRACE OPTION 2 AGAIN,
AND NOW, FOR THE LEFT BOUND OF THIS X INTERCEPT,
WE'RE GOING TO BE ABOVE THE X INTERCEPT.
SO WE'RE GOING TO PRESS THE RIGHT ARROW
UNTIL WE SEE THE CURSOR JUST ABOVE THAT X INTERCEPT
IN THE MIDDLE.
THERE IT IS.
PRESS ENTER.
PRESS THE RIGHT ARROW FOR THE RIGHT BOUND
AND THE RIGHT OF THIS X INTERCEPT MEANS BELOW,
BECAUSE ON THIS INTERVAL THE FUNCTION IS DECREASING.
PRESS ENTER, AND THEN WHEN IT SAYS, "GUESS,"
WE PRESS ENTER AGAIN.
SO NOTICE THE 0 IS X = 1.
AGAIN, THIS EXAMPLE IS PRETTY STRAIGHTFORWARD,
BUT WE CAN ALMOST TELL THE ZEROS JUST BY LOOKING AT THE GRAPH
RATHER THAN CALCULATING THEM,
BUT LET'S GO AHEAD AND FIND THIS LAST ONE.
AGAIN, 2nd TRACE OPTION 2.
PRESS THE RIGHT ARROW UNTIL WE'RE JUST BELOW
THE LAST X INTERCEPT.
NOTICE POINTS BELOW THIS ARE TO THE LEFT FOR LEFT BOUND.
PRESS ENTER.
FOR RIGHT BOUND WE NEED TO BE ABOVE THIS X INTERCEPT.
FOR EXAMPLE, HERE.
PRESS ENTER AND THEN ENTER AGAIN.
SO WE HAVE A 0 OF X = 5.
SO ALL THREE OF THE ZEROS ARE REAL AND RATIONAL,
AND WE FOUND ALL OF THEM USING THE TI84 GRAPHING CALCULATOR.
NOW, THERE IS ONE MORE CONNECTION I'D LIKE TO MAKE.
HAVING THE ZEROS OF THIS FUNCTION ALSO GIVE US
INFORMATION ABOUT THE FACTORS OF THE ORIGINAL FUNCTION.
WHAT I MEAN BY THAT IS IF X = -3 IS A 0 OF THIS FUNCTION,
THEN X + 3 MUST BE A FACTOR OF THE FUNCTION.
NOTICE HOW THE 0 IS -3.
THE FACTOR IS X + 3.
IF X = 1 IS A 0 OF THE FUNCTION,
THEN X - 1 MUST BE A FACTOR OF THE FUNCTION.
NOTICE WE PLACED 1 IN HERE.
THIS = 0, AND IF X = 5 IS A 0,
THEN X - 5 MUST ALSO BE A FACTOR.
IF WE MULTIPLIED THESE TOGETHER,
WE'D HAVE A LEADING TERM OF X CUBED.
THEREFORE, IF WE MULTIPLIED ALL THIS OUT,
WE WOULD END UP WITH THE ORIGINAL FUNCTION.
IN THE NEXT VIDEO WE'LL TAKE A LOOK
AT OUR EXAMPLE THAT'S A LITTLE BIT MORE CHALLENGING,
MEANING THE ZEROS WILL BE FRACTIONS.
I HOPE THIS WAS HELPFUL.