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- WE WANT TO FIND A POSSIBLE EQUATION
FOR A POLYNOMIAL FUNCTION
WITH REAL COEFFICIENTS HAVING ZEROS -4, -1, AND +3.
LET'S REVIEW WHAT WE KNOW ABOUT THE ZEROS
OF A POLYNOMIAL FUNCTION.
FIRST THE ZEROS OR ROOTS OF A POLYNOMIAL FUNCTION
ARE THE X VALUES FOR WHICH F OF X IS = TO 0.
SO THE VALUES OF X THAT MAKE THE FUNCTION = TO 0.
IF THE ZEROS OR ROOTS ARE R SUB 1, R SUB 2, AND SO-ON,
THIS GIVES US INFORMATION ABOUT FACTORS
OF THE POLYNOMIAL FUNCTION.
IF R SUB 1 IS A 0,
THEN X - R SUB 1 WOULD HAVE TO BE A FACTOR OF THE FUNCTION.
IF R SUB 2 IS A 0,
THEN X - R SUB 2 WOULD ALSO BE A FACTOR
OF THE POLYNOMIAL FUNCTION AND SO-ON.
NOTICE HOW THERE'S ALSO THIS A OUT HERE.
A CAN BE ANY CONSTANT.
NOTICE HOW IT'S NOT GOING TO AFFECT
THE ZEROS OF THE FUNCTION.
HOWEVER -- = 1 UNLESS R IS A FRACTION.
IF THE POLYNOMIAL FUNCTION HAS COMPLEX ZEROS OR ROOTS,
THESE ALWAYS COME IN CONJUGATE PAIRS.
AND THEN LASTLY, GRAPHICALLY, IF WE HAVE REAL ZEROS,
THE REAL ZEROS WOULD BE THE X INTERCEPTS
OF THE GRAPH OF THE FUNCTION.
SO GOING BACK TO OUR PROBLEM, IF WE KNOW THESE ARE OUR ZEROS,
THIS GIVES US INFORMATION ABOUT THE FACTORS
OF A POSSIBLE FUNCTION.
SO F OF X, A FUNCTION, WOULD HAVE TO HAVE A FACTOR OF X - -4
IF -4 IS A ZERO.
IF -1 IS A ZERO OUR FUNCTION MUST HAVE A FACTOR OF X - -1.
AND IF +3 IS A ZERO THEN WE'D HAVE TO HAVE A FACTOR OF X - 3.
AGAIN, WE COULD PUT A CONSTANT HERE.
SINCE WE DON'T HAVE ANY FRACTIONS,
WE'LL GO AHEAD AND LET A = 1.
SO NOW WE'RE GOING TO SIMPLIFY OUR BINOMIAL FACTORS
AND THEN FIND THIS PRODUCT.
SO F OF X IS GOING TO BE = TO--
WE HAVE A FACTOR OF X + 4, FACTOR OF X + 1,
AND A FACTOR OF X - 3.
NOW, TO MULTIPLY THIS OUT, THERE'S NO SHORTCUT,
WE CAN ONLY MULTIPLY TWO BINOMIALS AT A TIME.
SO WE'LL MULTIPLY THE QUANTITY X + 4 x THE QUANTITY X + 1,
SO WE'LL HAVE FOUR PRODUCTS,
1, 2, 3, AND 4.
SO WE'LL HAVE F OF X = X x X THAT'S X SQUARED.
X x 1 THAT'S 1X, AND THEN WE HAVE 4 x X THAT'S 4X.
1X + 4X IS 5X, THEN 4 x 1 IS +4.
SO WE HAVE THIS TRINOMIAL x THE BINOMIAL X - 3.
MULTIPLYING THESE TWO WE'LL HAVE SIX PRODUCTS,
ONE, TWO, THREE, FOUR, FIVE, AND SIX.
SO WE'LL HAVE F OF X EQUALS--
X TO THE SECOND x X THAT'S X TO THE THIRD.
X TO THE SECOND x -3 THAT'S -3X SQUARED OR -3X SQUARED.
THEN WE'LL HAVE 5X x X THAT'S 5X SQUARED, SO + 5X SQUARED.
5X x -3 THAT'S -15X OR -15X,
AND THEN WE HAVE 4 x X THAT'S + 4X.
THEN WE HAVE 4 x -3 THAT'S -12, SO WE HAVE -12.
LAST STEP, WE WANT TO COMBINE OUR LIKE TERMS.
SO WE HAVE TWO X SQUARED TERMS, AND WE HAVE TWO X TERMS.
SO ONE POSSIBLE FUNCTION WITH THESE ZEROS WOULD BE X CUBED,
-3X SQUARED + 5X SQUARED THAT'S +2X SQUARED,
SO +2X SQUARED.
-15X + 4X IS -11X OR -11X
AND THEN -12.
SO, AGAIN, THIS POLYNOMIAL FUNCTION,
WHICH IS ALSO A CUBIC FUNCTION BECAUSE IT'S DEGREE 3,
WOULD HAVE THE REAL ZEROS OF -4, -1, AND +3.
AND TO VERIFY THIS, SINCE THESE ARE REAL ZEROS,
WE CAN GRAPH THIS FUNCTION
AND VERIFY THESE WOULD BE THE X INTERCEPTS OF THE GRAPH.
SO HERE'S A GRAPH OF THE FUNCTION THAT WE JUST FOUND.
NOTICE HOW WE HAVE AN X INTERCEPT OF -4, -1, AND +3,
WHICH VERIFIES OUR WORK.
OKAY, I HOPE YOU FOUND THIS HELPFUL.