Polynomial 1.9.9.4.

Number Theory Level 3

P ( x ) P(x) is a polynomial with integral coefficients such that the absolute value of the constant term of P ( x ) P(x) is smaller than 1000. Given further that P ( 19 ) = P ( 94 ) = 1994 P(19)=P(94)=1994 , find the constant term of the polynomial P ( x ) P(x) .


The answer is 208.

View wiki
Ellen Shang by Ellen Shang , 23, Canada

This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try refreshing the page, (b) enabling javascript if it is disabled on your browser and, finally, (c) loading the non-javascript version of this page . We're sorry about the hassle.

4 solutions

Lu Chee Ket
Jan 30, 2015

P ( x ) = a ( x − 19 ) ( x − 94 ) + 1994 P(x) = a(x - 19)(x - 94) + 1994

If x = 0 x = 0 , then a ( − 19 ) ( − 94 ) + 1994 = 1786 a + 1994 a(-19)(-94) + 1994 = 1786a + 1994

For constant term of absolute value smaller than 1000, then a a must be − 1 -1 .

Therefore, 1786(-1) + 1994 = 208.

13 Helpful 1 Interesting 1 Brilliant 1 Confused

But how can u consider p(x) to be a quadratic function. There can be a million functions which satisfy the above conditions. For example 1: P(x)= (x-19)(x-94)(x-1) + 1994 and 2: P(x)=(x-19)(x-94)(x-1)(x-2) + 1994

Arnab Das - 6 years, 4 months ago

Log in to reply

You're right. But if P ( x ) = h ( x ) ( x − 19 ) ( x − 94 ) + 1994 P(x) = h(x) (x-19)(x-94)+1994 , the constant term of P ( x ) P(x) will be 1786 h ( 0 ) + 1994 1786h(0) + 1994 , so h ( 0 ) = − 1 h(0) = -1 is forced and the constant term of P ( x ) P(x) must be 208 208 . But yes, h ( x ) h(x) can be any polynomial whose constant term is − 1 -1 .

Patrick Corn - 6 years, 4 months ago
Shabnam Lohawala
Feb 5, 2015

Let the constant term be 'a'. Since P(19)=1994 ,so (1994 - a) is divisible by 19. P(94)=a,so (1994 - a) is divisible by 94. (1994 - a ) is divisible by 19×94 = 1786. Since |a| is less than 1000, (1994 -a) must lie between 994 and 2994. So (1994 -a) = 1786 and a=208.

6 Helpful 0 Interesting 0 Brilliant 0 Confused
Cantdo Math
Apr 18, 2020

p(x)=18 modulo 19 and p(x)=20 modulo 98,gives p(x)=208 modulo 1786.Now,|constant term|<1000 implies 208 is indeed the answer.

0 Helpful 0 Interesting 0 Brilliant 0 Confused
Incredible Mind
Jan 30, 2015

the polynomial is k(x-19)(x-94) + 1994 where k is integer.

Next constant term of polynomial is

k(19)94

Also k(19)94<1000

It seems answer is 208 that is k=-1. But i have no idea why k = -1 OR A/C k<= - 1 should work.I believe the question also has the condition constant term>0

0 Helpful 0 Interesting 0 Brilliant 0 Confused

But how can u consider p(x) to be a quadratic function. There can be a million functions which satisfy the above conditions. For example 1: P(x)= (x-19)(x-94)(x-1) + 1994 and 2: P(x)=(x-19)(x-94)(x-1)(x-2) + 1994

Arnab Das - 6 years, 4 months ago

the questions says, absolute value of constant term is less than 1000. The constant term is k(19)(94)+1994, not just k(19)(94)

Murali Kancharla - 6 years, 4 months ago

0 pending reports

×

Problem Loading...

Note Loading...

Set Loading...