Integers as Roots of Polynomial: Find Possible Values | POTW #293 Dec 19th, 2017

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In summary, integers as roots of a polynomial are values that result in a zero answer when substituted into a polynomial equation. To find these values, the rational roots theorem and the synthetic division method can be used. It is important to find these values as they help solve equations and understand the behavior of the polynomial. However, there are limitations to finding these values, such as only applying to polynomials with rational coefficients and not guaranteeing all possible roots will be found. Additionally, a polynomial can have more than one set of possible integer roots, but only a maximum of n roots, where n is the degree of the polynomial.
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anemone
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Here is this week's POTW:

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Suppose $n \ge 0$ and all the roots of $x^3+ax+4-(2\times 2016^n)=0$ are integers. Find all possible values of $a$.

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Remember to read the http://www.mathhelpboards.com/showthread.php?772-Problem-of-the-Week-%28POTW%29-Procedure-and-Guidelines to find out how to http://www.mathhelpboards.com/forms.php?do=form&fid=2!
 
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Hi MHB!

I want to apologize for the $n$ for this week problem should be always greater than or equal to zero, rather than it is a positive integer and I have made the respective change to above post.

I am truly sorry for making this typo and I hope this clears some members confusion and I am looking forward to receive more submissions to this week problem!
 
  • #3
Congratulations to castor28 for his correct solution! (Smile)

Suggested solution:
Let the integer roots of the equation be $m,\,n$ and $k$. Then by Vieta's formulas, we have

$m+n+k=0,\,mn+nk+km=a,\,mnk=2\times 2016^n-4$.

Assume that $n\ge 1$, then $7 \nmid m,\,n,\,k$ and from $(m+n+k)^3=0$ we get

$m^3+n^3+k^3=3mnk\equiv 2(\mod 7)$

Note that the non-zero cubic residues modulo 7 are 1 and 6 and so we cannot have $m^3+n^3+k^3=3mnk\equiv 2(\mod 7)$.

Thus $n=0$, $\therefore m+n+k=0$ and $mnk=-2$, which gives $(m,\,n,\,k)=(1,\,1,\,-2)$ and its cyclic permutations.

Therefore $a=-3$.
 

FAQ: Integers as Roots of Polynomial: Find Possible Values | POTW #293 Dec 19th, 2017

1. What are integers as roots of a polynomial?

Integers as roots of a polynomial refer to the values that, when substituted into a polynomial equation, result in an answer of zero. These values are also known as "solutions" or "zeros" of the polynomial.

2. How do you find the possible values for integers as roots of a polynomial?

To find the possible values for integers as roots of a polynomial, you can use the rational roots theorem and the synthetic division method. The rational roots theorem states that the possible rational roots of a polynomial are all the factors of the constant term divided by all the factors of the leading coefficient. The synthetic division method is then used to test these possible roots and find the actual roots of the polynomial.

3. Why is it important to find the possible values for integers as roots of a polynomial?

Finding the possible values for integers as roots of a polynomial is important because it allows us to solve polynomial equations and find the solutions to real-life problems. These values can also help us understand the behavior of the polynomial, such as its intercepts and turning points.

4. Are there any limitations to finding the possible values for integers as roots of a polynomial?

Yes, there are limitations to finding the possible values for integers as roots of a polynomial. The rational roots theorem only applies to polynomials with rational coefficients, meaning that all the coefficients in the polynomial must be whole numbers or fractions. It also does not guarantee that all the possible roots will be found, as there may be irrational or imaginary roots.

5. Can a polynomial have more than one set of possible values for integers as roots?

Yes, a polynomial can have more than one set of possible values for integers as roots. This is because a polynomial can have multiple factors and each factor can have its own set of possible roots. However, a polynomial can only have a maximum of n possible integer roots, where n is the degree of the polynomial.

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