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- WE WANT TO FIND AN EQUATION FOR THE GRAPH OF THE DEGREE 6 POLYNOMIAL FUNCTION AND ALSO LEAVE THE FUNCTION IN FACTORED FORM. BECAUSE THE POLYNOMIAL IS DEGREE 6 WE KNOW THERE IS AT MOST 6 REAL RATIONAL ZEROS OR ROOTS. AND GRAPHICALLY, WE CAN FIND THESE BY ANALYZING THE X INTERCEPTS OF OUR GRAPH. SO LOOKING AT THE GRAPH NOTICE HOW WE HAVE 1, 2, 3, 4 X INTERCEPTS, THEREFORE THERE ARE FOUR REAL RATIONAL ZEROS OR ROOTS BUT BECAUSE HOW THE FUNCTION BEHAVES WE CAN SEE THAT SOME OF THE ROOTS HAVE A MULTIPLICITY THATS GREATER THAN 1. WHAT I MEAN BY THAT IS IF A FUNCTION CROSSES THE X AXIS THEN THE ROOT OR ZERO HAS AN ODD MULTIPLICITY. IF IT TOUCHES THE X AXIS BUT DOESNT CROSS THE X AXIS THE MULTIPLICITY IS EVEN. SO NOTICE THE ZEROS AT (-2, +5) DO JUST TOUCH THE X AXIS BUT DONT CROSS THE X AXIS WHICH MEANS THE ZEROS OF (-2, +5) HAVE AN EVEN MULTIPLICITY AND BECAUSE WE HAVE FOUR X INTERCEPTS WE COULD ASSUME BOTH HAVE A MULTIPLICITY OF 2. SO IF THESE TWO HAVE MULTIPLICITY OF 2, 2, 4, 5, 6 WE
