A polynomial with integer coefficients is of the form \(x^4 + a_3 x^3 + a_2 x^2 + a_1 x + 18.\)You are told that the integer r is a double root of this polynomial. (In other words, the polynomial is divisible by (x - r)^2.) Enter all the possible values of r, separated by commas.

Guest May 24, 2019

#1**+2 **

I think that the possible roots are

1 and -1 up to 4 times

2 and -2 once

3 an -3 up to twice - becasue 9 goes into 18

9 and -9 once

18 and -18 once.

So I think that the possible values of r are, -3, -1, 1, 3

Melody May 24, 2019

#2**+1 **

I agree with Melody

Let m and n be the other two roots

This would imply that r^2 * m * n = 18 ⇒ m * n = 18 / r^2

Using the Rational Zeroes Theorem, the divisors of 18 are

± [ 1,2, 3, 6, 9 , 18 ]

If we have integer coefficients, then m and n must be integers themselves....and their product must also be an integer

But this can only be true if r = ±1 or r = ±3

CPhill May 24, 2019