Let $P(x) = x^3 - 5 x^2 + 7 x + k = 0$. If P(x) has a double root in the set of integers, find the value of k using Vieta's Formulas.

Question

I managed to derive Vieta's Formulas intuitively to find the values for each coefficient of $n$ degree of $x$ but didn't have success in finding the value of $k$. I figured out it was simultaneous equations however still couldn't find the value of $k$ being the constant in $P(x)$. Help would be great as questions like this will be in my year $10$ yearly exam.

P.S, please answer this question in reference to Vieta's Formulas and polynomial terminology, no calculus :) thanks.

score 1 · Accepted Answer · answered Dec 05 '20 at 04:38

1

Let the roots be $r, r, s$. Vieta's formulas give the following system: $$2r+s=5\\ r^2+2rs=7\\ r^2s=-k$$

Substituting the first equation into the second, $$r^2+2r(5-2r)=7.$$

Can you take it from here?

answered Dec 05 '20 at 04:38

Théophile

24,627

Lol biggest mathematical blunder of my life, Thanks – Great Leader Dec 10 '20 at 00:33

Let $P(x) = x^3 - 5 x^2 + 7 x + k = 0$. If P(x) has a double root in the set of integers, find the value of k using Vieta's Formulas.

1 Answers1