Polynomials are not fun

Algebra Level pending

${x,y,z}$ are numbers such that $xyz=1$ , $xy+xz+yz=2$ , and $x+y+z=3$ . If $x,y$ and $z$ are the roots of the cubic polynomial $3n^3-an^2+bn-c=0$ , what is $a+b+c$ ?

The answer is 18.

1 solution

Tom Engelsman
Sep 18, 2017

For the above polynomial, we have:

$n^3 - \frac{a}{3} n^2 + \frac{b}{3} n - \frac{c}{3} = (n-x)(n-y)(n-z) = 0$ .

By Vieta's Formulae, we can deduce:

$\frac{a}{3} = x + y + z = 3 \Rightarrow \boxed{a = 9};$

$\frac{b}{3} = xy + xz + yz = 2 \Rightarrow \boxed{b = 6};$

$\frac{c}{3} = xyz = 1 \Rightarrow \boxed{c = 3}.$

Hence, $a + b + c = 9 + 6 + 3 =\boxed{18}.$

Polynomials are not fun

The answer is 18.

1 solution

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