In Exercises 33–38, use Descartes's Rule of Signs to determine th...

Question

In Exercises 33–38, use Descartes's Rule of Signs to determine the possible number of positive and negative real zeros for each given function. f(x)=x^3+2x^2+5x+4

Accepted Answer

Hey everyone here we are asked to use the cards rule of signs to identify the possible number of positive and negative real zeros of the function F. Of X is equal to three X cubed plus four X squared plus seven X plus one. Now to start with the possible number of positive real zeros. We need to make sure we have a positive X input which we do and it is given by our given function. So from here we can just start noting the number of sign changes. And with our positive input function here we start with a positive three X cubed. And then we go to another positive term of four X squared. And then we go to another positive of seven X. And also another positive of one. And therefore we can see we have zero sign changes. And from there we conclude that the number of positive real zeros is also zero. We don't have any positive real zeros. And so now we repeat this process. But for the negative possible real zeros we have to make sure that the X input is negative. So we take F of negative X. Which gives us three times the quantity of negative X Q plus four times the quantity of negative X squared plus seven times the quantity of negative X plus one. And this simplifies to negative three X Q plus four X squared minus seven X plus one. And so now once again we need to note the number of sign changes. And so for our negative input we can see that we go from a negative to a positive. So there is one sign change here and then we go from positive to negative. So this is our second sign change and then we end by going negative to positive, which would be our third sign change. And therefore we have a total of three sign changes. And from this we can conclude that the number of negative possible real zeros would be three or one being three hour maximum here. And with this we know that we have answer choice D. Where we have no positive real zeros and three or one negative real zeros. Thanks for watching. I hope you found this video helpful and I'll see you again next time.

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In Exercises 33–38, use Descartes's Rule of Signs to determine the possible number of positive and negative real zeros for each given function. f(x)=x^3+2x^2+5x+4