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# Help

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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.

May 24, 2019

#1
+106885
+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

May 24, 2019
#2
+106514
+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

May 24, 2019
#3
+106885
+1

Thanks Chris

Melody  May 25, 2019