Can a Polynomial Equation Have Roots in Arithmetic Progression?

[anemone]

Find $a$ and $b$ such that the equation $x^6+ax^4+bx^2-225=0$ has six real roots in arithmetic progression.

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[anemone]

Congratulations to the following members for their correct solutions::)

1. MarkFL
2. kaliprasad
3. greg1313
4. Fallen Angel
5. lfdahl
6. chisigma

Solution from kaliprasad:

Spoiler

Because of coefficient of $x^5 = 0$, hence the sum of the six roots is zero.

Let the roots be $-5a,-3a,-a,a,3a,5a$ with $a \gt 0$.

$\text{product of roots} = -225a^6 = -225$ or $a= 1$

so roots are -5,-3,-1,1,3,5.

Given the polynomial is

$(x+5)(x+3)(x+1)(x-1)(x-3)(x-5) = (x^2-25)(x^2-9)(x^2-1)$

=$(x^6 - (25+9+1)x^4 + ( 25 * 9 + 25 * 1 +9) x^2 - 225$

= $x^6 - 35x^4 + 259x^2-225$

so a = - 35 and b = 259

Solution from Fallen Angel:

Spoiler

First of all, we can change $y=x^2$ and get the polynomial $P(y)=y^3+a^2+by-225$

Now if $\alpha ,\beta ,\gamma$ are the roots of $P$, the condition of the roots of the original polynomial being in arithmetic progression implies $\sqrt{\beta}=3\sqrt{\alpha}$ and $\sqrt{\gamma}=5\sqrt{\alpha}$

Now we can factor $P(y)=(y-\alpha)(y-9\alpha)(y-25\alpha)$ and we obtain $\alpha=1$, $a=-35$, $b=259$.

Back to the original polynomial we got that the roots of $x^6-35x^4+259x^2-225$ are $-5,-3,-1,1,3,5$.

Solution from lfdahl:

Spoiler

Given $P(x) = x^6+ax^4+bx^2-225 = 0$.

$P(x) = P(-x)$ so $P$ is an even function. Therefore, the roots are symmetrically distributed around $x = 0$.

If I let $r$ denote the smallest positive root (and $-r$ the largest negative) I need to factorize $P(x)$ such that the numerical distance between neighbouring roots is $2r$: $\pm r, \pm 3r, \pm 5r$, and $P(x)$ has the form:

$P(x) = (x+r)(x-r)(x+3r)(x-3r)(x+5r)(x-5r)= (x^2-r^2)(x^2-9r^2)(x^2-25r^2) $.

The root product $(-r^2)(-9r^2)(-25r^2)$ should be $-225$: $-9\cdot25r^6 = -225$ so $r = \pm 1$.

Thus the factorization is simply: $P(x) = (x^2-1)(x^2-9)(x^2-25) = x^6-35x^4+259x^2-225$.

The coefficients $a$ and $b$ must therefore be: $a=-35$ and $b = 259$.