Half way squared polynomials

Algebra Level 5

If $P(x)=x^{3}-3x^{2}+2x+5$ and $a_{1}, a_{2}, a_{3}$ are roots of $P(x)$ , then find the sum of the coefficients of a monic polynomial whose roots are $(a_{1})^{2}a_{2}a_{3},(a_{2})^{2}a_{1}a_{3},(a_{3})^{2}a_{2}a_{1}$ .

The answer is -559.

2 solutions

Alan Enrique Ontiveros Salazar
Aug 15, 2015

Let $a_1=a, a_2=b, a_3=c$ . By Vieta's formulas, $a+b+c=3$ , $ab+ac+bc=2$ and $abc=-5$ .

So the polynomial we want is

$Q(x)=(x-a^2bc)(x-ab^2c)(x-abc^2)\\ Q(x)=x^3-abc(a+b+c)x^2+(abc)^2(ab+ac+bc)x-(abc)^4 \\ Q(x)=x^3+15x^2+50x-625$

Hence the sum of the coefficients is $\boxed{-559}$ .

I added the coefficients to get -560 and didn't consider the leading 1... lol. Tricky question.

Jonathan Hocker - 5 years, 7 months ago

Xuming Liang
Aug 15, 2015

Note that $a_1a_2a_3=-5$ , then a polynomial that has roots of those desired forms is $P(-\frac {x}{5})$ . Making this polynomial monic and adding the coefficients gives us the answer of $\boxed{-559}$

Half way squared polynomials

The answer is -559.

2 solutions

0 pending reports