GMAT Question of the Day - Daily to your Mailbox; hard ones only

 It is currently 13 Dec 2019, 06:04 ### GMAT Club Daily Prep

#### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History

#### Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.  # Neat Fact for Integral Solutions to a polynomial

Author Message
TAGS:

### Hide Tags

Verbal Forum Moderator B
Joined: 10 Oct 2012
Posts: 583
Neat Fact for Integral Solutions to a polynomial  [#permalink]

### Show Tags

5
14
Hello!

Consider any polynomial $$f(x) = A_1x^n+A_2x^{n-1}+.....A_n$$

Assumption : All the co-efficients for the given polynomial have to be integral,i.e. $$A_1,A_2,A_3....A_n$$ are all integers.

Fact:Any integral solution(root) for the above polynomial will always be a factor(positive/negative) of the constant term : $$A_n$$

Example I : $$f(x) = 5x^2-16x+3$$. Thus, we know that if the given polynomial has any integral solutions, then it will always be out of one of the following : $$-3,-1,1,3$$

We see that only x=3 is a root for the given polynomial. Also, we know that product of the roots is$$\frac {3}{5}$$. Hence, the other root is $$\frac {1}{5}$$

Example II : Find the no of integral solutions for the expression $$f(x) = 3x^4-10x^2+7x+1$$

A. 0
B. 1
C. 2
D. 3
E. 4

For the given expression, instead of finding the possible integral solutions by hit and trial, we can be rest assured that if there is any integral solution, it will be a factor of the constant term ,i.e. 1 or -1. Just plug-in both the values, and we find that f(1) and f(-1) are both not equal to zero. Thus, there is NO integral solution possible for the given expression--> Option A.

Example III : Find the no of integral solutions for the expression $$f(x) = 4x^4-8x^3+9x-3$$

A. 0
B. 1
C. 2
D. 3
E. 4

Just as above, the integral roots of the given expression would be one of the following : -3,-1,1,3. We can easily see that only x = -1 satisfies. Thus, there is only one integral solution for the given polynomial-->Option B.

Hence, keeping this fact in mind might just reduce the range of the hit and trial values we end up considering.
_________________
Non-Human User Joined: 09 Sep 2013
Posts: 13742
Re: Neat Fact for Integral Solutions to a polynomial  [#permalink]

### Show Tags

Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.
_________________ Re: Neat Fact for Integral Solutions to a polynomial   [#permalink] 09 Aug 2018, 11:48
Display posts from previous: Sort by

# Neat Fact for Integral Solutions to a polynomial  