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.
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
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