# Solving 8 Roots: Can 3 Quadratic Polynomials Fulfill $f(g(h(x)))=0$?

• MHB
• anemone
In summary, it is possible for 3 quadratic polynomials to fulfill the equation $f(g(h(x)))=0$ by constructing a scenario with specific coefficients and values. The general approach to solving for 8 roots involves breaking down the equation into individual quadratic polynomials and solving for their roots separately. There are some algebraic shortcuts that can be used to simplify the equation, and the order in which the polynomials are evaluated does not matter. A computer program can also be used to solve this equation, which may be more efficient and accurate for complex coefficients or a large number of roots.
anemone
Gold Member
MHB
POTW Director
Is it possible to find three quadratic polynomials $f(x),\,g(x)$ and $h(x)$ such that the equation $f(g(h(x)))=0$ has the eight roots 1, 2, 3, 4, 5, 6, 7 and 8?

Suppose there are such $f,\,g,\,h$. Then $h(1),\,h(2),\cdots,\,h(8)$ will be the roots of the 4th degree polynomial $f(g(x))$. Since $h(a)=h(b),\,a\ne b$ if and only if $a$ and $b$ are symmetric with respect to the axis of the parabola, it follows that $h(1)=h(8),\,h(2)=h(7),\,h(3)=h(6),\,h(4)=h(5)$ and the parabola $y=h(x)$ is symmetric with respect to $x=\dfrac{9}{2}$. Also, we have either $h(1)<h(2)<h(3)<h(4)$ or $h(1)>h(2)>h(3)>h(4)$.

Now, $g(h(1)),\,g(h(2)),\,g(h(3)),\,g(h(4))$ are the roots of the quadratic polynomial $f(x)$, so $g(h(1))=g(h(4))$ and $g(h(2))=g(h(3))$ , which implies $h(1)+h(4)=h(2)+h(3)$. For $h(x)=Ax^2+Bx+C$, this would force $A=0$, a contradiction.

## 1. How do you solve for 8 roots in a polynomial equation?

To solve for 8 roots in a polynomial equation, you will need to use a combination of algebraic techniques such as factoring, the quadratic formula, and synthetic division. You may also need to use the rational root theorem and the remainder theorem to find all possible roots.

## 2. Can 3 quadratic polynomials fulfill the equation $f(g(h(x)))=0$?

Yes, it is possible for 3 quadratic polynomials to fulfill the equation $f(g(h(x)))=0$. This can be achieved by composing the three polynomials in a specific order, where the output of one polynomial becomes the input of the next.

## 3. What is the importance of solving for 8 roots in a polynomial equation?

Solving for 8 roots in a polynomial equation allows us to find all possible solutions to the equation. This is important in various fields of science and mathematics, as it helps us understand the behavior and patterns of the equation and its corresponding graph.

## 4. Are there any shortcuts or tricks for solving for 8 roots in a polynomial equation?

There are no set shortcuts or tricks for solving for 8 roots in a polynomial equation. However, having a strong understanding of algebraic techniques and being able to recognize patterns in the equation can make the process easier and more efficient.

## 5. Can technology be used to solve for 8 roots in a polynomial equation?

Yes, technology such as graphing calculators or computer software can be used to solve for 8 roots in a polynomial equation. However, it is important to have a basic understanding of the algebraic techniques involved in order to interpret and verify the results given by technology.

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