Precalculus : Descartes' Rule, Intermediate Value Theorem, Sum and Product of Zeros
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Example Questions
Example Question #1 : Find The Sum And Product Of The Zeros Of A Polynomial
Given 
To determine zeros of 
Set each of the factorized components equal to zero and solve for 
The sum of the roots:
The product of the roots:
Example Question #2 : Find The Sum And Product Of The Zeros Of A Polynomial
Please choose the best answer from the following choices.
Find the sum and product of the zeros of the following polynomial:
To find the zeros you have to factor the polynomial.
This is easily factorable and you will get 
Next, set both of these equal to zero. 
Isolate the x's and you will get 
The sum will be 
Example Question #3 : Find The Sum And Product Of The Zeros Of A Polynomial
Please choose the best answer from the following choices.
Find the sum and product of the zeros of the following polynomial:
Factoring the polynomial will give you 
The zeros of these binomials will be 
If you add these together (sum) you get 
Example Question #1 : Determine The Number Of Positive And Negative Real Zeros Of A Polynomial Using Descartes' Rule Of Signs
Determine the possible number of Positive and Negative Real Zeros of the polynomial using Descartes' Rule of Signs
Positive zeroes:
Negative zeroes:
Positive zeroes:
Negative zeroes:
Positive zeroes:
Negative zeroes:
Positive zeroes:
Negative zeroes:
Positive zeroes:
Negative zeroes:
In order to determine the positive number of real zeroes, we must count the number of sign changes in the coefficients of the terms of the polynomial. The number of real zeroes can then be any positive difference of that number and a positive multiple of two.
For the function
there are four sign changes. As such, the number of positive real zeroes can be
In order to determine the positive number of real zeroes, we must count the number of sign changes in the coefficients of the terms of the polynomial after substituting for 
After substituting 
and there is one sign change.
As such, there can only be one negative root.
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