Question 580756: Make up a polynomial that has the following characteristics: crosses the x-axis at -1 and 4, touches the x-axis at 0 and 2, and is above the x-axis between 0 and 2.
Answer by KMST(5328)   (Show Source):
You can put this solution on YOUR website! If  , or  , or , or ... (well, you get the idea) is a factor of a polynomial, the polynomial will be zero AND change sign at x=4. Then the polynomial "crosses the x-axis at ... 4". To make it the simplest, we should include
 as a factor in the polynomial, so that it crosses the x-axis at -1 and 4."
On the other hand if we include  and  as factors in the polynomial, the polynomial will be zero, but will not change sign at x=0 and x=2, so we would say that it touches (but does not cross) the x-axis.
So  would be a polynomial that "crosses the x-axis at -1 and 4," and "touches the x-axis at 0 and 2."
How does it behave between 0 and 2? If it is positive, then it "is above the x-axis between 0 and 2."
But  , so we need a (-1) as an extra factor.
 is a polynomial that complies with the problem's requirements.
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