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Finding a Polynomial Function, find a polynomial function that has the given

zeros. (There are many correct answers.)

$$

0,1,10

$$

$f(x)=x(x-1)(x-10)=x^3-10x^2-x^2+10x=x^3-11x^2+10x$

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Missouri State University

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University of Michigan - Ann Arbor

we're going to look for a polynomial function that has the given zeros. Now, the relationship in this question that's important is the relationship between zeros and the factors of a polynomial equation. So we're essentially going to be working backwards a little bit Now when we have a zero of zero, This would imply the factor X zero, which is really just the same thing as X. When we have the zero of one, that would give us the factor X -1. Because if you did plug a one into that 1 -1 would make zero. So there's our second factor And last for the zero of 10. This would imply a factor of x minus 10 because when plugging a 10 into that expression, 10 minus 10 would make zero. Therefore 10 would be a zero as well, Putting all of these together in the form of one polynomial, we'll call it a five ax. And this function would have the form X times the quantity X -1 Times The Quantity X -10. It's very important to note that in this question, we know there are multiple answers. The X the x minus one and the x minus 10 are non negotiable. Those have to be part of it. But any multiple of this would also work. So for example, if we placed a five in front multiply everything by five, that would work. If we plugged a foreign front, that would be okay. That would still have those same zeros. So multiplying by some constant wouldn't change those factors we have but just to kind of keep it in the formative, we will distribute it out. So we do have it in standard form. The first distributing will do will give us x squared minus x. And then in turn we would have to multiply that quantity times X -10. And when we distribute that out we would get x squared times X, which is X to the third power or x cubed. We have x squared minus 10 Which is -10 x squared. We would have minus x times X, which we minus X squared. And finally a negative X times negative 10 which would give us plus 10 X. Now we do have to like terms in here the minus 10 X squared in the minus X squared. So we'll write this up above in blue, it's final simplified form, Our polynomial. That would have those zeros of 0.1 intent would be X cubed -11 x squared Plus 10 x. and again any any constant that would get multiplied by. That polynomial would give you additional multiple answers.