Max Value of $b$ for Real Roots of $f(x)$ and $g(x)$

In summary, the maximum value of $b$ for real roots of $f(x)$ and $g(x)$ is dependent on the coefficients of the polynomials and can be calculated by finding the discriminant of the quadratic equation formed by setting them equal to 0. To determine if $f(x)$ and $g(x)$ have real roots, the discriminant must be greater than or equal to 0. However, there are limitations to finding the maximum value of $b$, such as only being able to calculate it for polynomials of degree 2 or lower and not being able to determine it if the coefficients are imaginary or complex numbers. Altering the coefficients can also change the maximum value of $b$. Additionally, the maximum value
  • #1
anemone
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Let $a$ and $b$ be real numbers and $r,\,s$ and $t$ be the roots of $f(x)=x^3+ax^2+bx-1$ and $g(x)=x^3+mx^2+nx+p$ has roots $r^2,\,s^2$ and $t^2$. If $g(-1)=-5$, find the maximum possible value of $b$.
 
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Rewrite $g(x)$ in terms of $a$ and $b$...
 
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From Vieta's formulas, we know

$=a=r+s+t,\,b=rs+st+tr,\,1=rst$

and

$m=-(r^2+s^2+t^2),\,n=r^2s^2+s^2t^2+t^2r^2,\,p=-r^2s^2t^2$

Let's try to express $m,\,n$ and $p$ in terms of $a$ and $b$. The easiest one is $p$:
$p=-(rst)^2=-1$

From $m$, we square $r+s+t$:
$a^2=(r+s+t)^2=r^2+s^2+t^2+2(rs+st+tr)=-m+2b\implies m=2b-a^2$

Finally, for $n$ we can square $rs+st+tr$:
$b^2=(rs+st+tr)^2=r^2s^2+s^2t^2+t^2r^2+2(r^2st+rs^2t+t^2rs)=n+2rst(r+s+t)=n-2a\implies n=b^2+2a$

So now we write $g(x)$ in terms of $a$ and $b$:
$g(x)=x^3+(2b-a^2)x^2+(b^2+2a)x-1$

We know that $g(-1)=-5$; when we plug this into our equation for $g(x)$ we get

$-5=(-1)^3+(2b-a^2)(-1)^2+(b^2+2a)(-1)-1=1b-a^2-b^2-2a-2\implies a^2+2a+b^2-2b-3=0$

We seek the largest possible value of $b$. Since $a$ is real, we know that the discriminant of this quadratic must be non-negative. In particular,

$2^2-4(b^2-2b-3)\ge 0\implies b^2-2b-4\le 0$

Solving this quadratic gives us that the largest possible value of $b$ is $1+\sqrt{5}$.
 

Related to Max Value of $b$ for Real Roots of $f(x)$ and $g(x)$

1. What is the significance of finding the maximum value of $b$ for real roots of $f(x)$ and $g(x)$?

Finding the maximum value of $b$ for real roots of $f(x)$ and $g(x)$ can help determine the range of values for which the equations have real solutions. This information is important in many fields of science, such as engineering and physics, where finding the roots of equations is necessary for solving problems.

2. How is the maximum value of $b$ calculated for real roots of $f(x)$ and $g(x)$?

The maximum value of $b$ can be calculated by setting the discriminant of the quadratic equations $f(x)$ and $g(x)$ to 0. This will result in a quadratic equation with one variable, which can then be solved using the quadratic formula. The resulting value of $b$ will be the maximum value for which the equations have real roots.

3. Can the maximum value of $b$ be negative for real roots of $f(x)$ and $g(x)$?

Yes, the maximum value of $b$ can be negative for real roots of $f(x)$ and $g(x)$. This occurs when the discriminant is negative, indicating that the equations have no real solutions. In this case, the maximum value of $b$ would be the value for which the discriminant is 0, resulting in a complex solution.

4. How does the maximum value of $b$ affect the number of real roots of $f(x)$ and $g(x)$?

The maximum value of $b$ can determine the number of real roots of $f(x)$ and $g(x)$. If the maximum value of $b$ is positive, the equations will have two real roots. If the maximum value of $b$ is 0, the equations will have one real root. And if the maximum value of $b$ is negative, the equations will have no real roots.

5. Are there any limitations to using the maximum value of $b$ for determining real roots of $f(x)$ and $g(x)$?

Yes, there are limitations to using the maximum value of $b$ for determining real roots of $f(x)$ and $g(x)$. This method only works for quadratic equations, and does not apply to higher degree polynomials. Additionally, the equations must be in standard form, with the highest degree term having a coefficient of 1.

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