How Do You Solve Complex Root Expressions in Polynomial Equations?

  • MHB
  • Thread starter anemone
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    2015
In summary, the purpose of finding the sum of expressions involving roots is to simplify and solve complex equations. These expressions can contain any type of root and can be combined with other operations. The general process for finding the sum involves simplifying each root expression, combining like terms, and rationalizing irrational roots. Common mistakes to avoid include forgetting to simplify, combining unlike terms, and not rationalizing. There are also shortcuts and tricks that can make the process easier, such as using the distributive property and recognizing patterns.
  • #1
anemone
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MHB
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Here is this week's POTW:

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Let $u,\,v,\,w$ be the roots of the equation $x^3-6x^2+18x-36=0$.

Evaluate

$\left(\dfrac{u}{v}+\dfrac{v}{u}+\dfrac{v}{w}+\dfrac{w}{v}+\dfrac{u}{w}+\dfrac{w}{u}+3\right)(3^{u^2+v^2+w^2})^{u^3+v^3+w^3}$.

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  • #2
No one answered last week High School's POTW correctly but an honorable mention goes to lfdahl, as he computed all values correct except for the value of $\left(\dfrac{u}{v}+\dfrac{v}{u}+\dfrac{v}{w}+\dfrac{w}{v}+\dfrac{u}{w}+\dfrac{w}{u}+3\right)$.

Here's the proposed solution:

We're told $u,\,v,\,w$ are the roots of the equation $x^3-6x^2+18x-36=0$.

Vieta's formulas tell us then that

$u+v+w=6$; $uv+uw+vw=18$; $uvw=36$

Not that the expression inside the first parentheses could be algebraically modified so we could determine its value:

$\begin{align*}\dfrac{u}{v}+\dfrac{v}{u}+\dfrac{v}{w}+\dfrac{w}{v}+\dfrac{u}{w}+\dfrac{w}{u}+3&=1+\dfrac{u}{v}+\dfrac{u}{w}+1+\dfrac{v}{u}+\dfrac{v}{w}+1+\dfrac{w}{v}+\dfrac{w}{u}\\&=\dfrac{u}{u}+\dfrac{u}{v}+\dfrac{u}{w}+\dfrac{v}{v}+\dfrac{v}{u}+\dfrac{v}{w}+\dfrac{w}{w}+\dfrac{w}{v}+\dfrac{w}{u}\\&=u\left(\dfrac{1}{u}+\dfrac{1}{v}+\dfrac{1}{w}\right)+v\left(\dfrac{1}{u}+\dfrac{1}{v}+\dfrac{1}{w}\right)+w\left(\dfrac{1}{u}+\dfrac{1}{v}+\dfrac{1}{w}\right)\\&=(u+v+w)\left(\dfrac{1}{u}+\dfrac{1}{v}+\dfrac{1}{w}\right)\\&=(u+v+w)\left(\dfrac{uv+uw+vw}{uvw}\right)\\&=(6)\left(\dfrac{18}{36}\right)\\&=3\end{align*}$

We also have

$u^2+v^2+w^2=(u+v+w)^2-2(uv+uw+vw)=6^2-2(18)=0$ and

$u^3+v^3+w^3=(u+v+w)^3-3(u+v+w)(uv+uw+vw)+3uvw=6^3-3(6)(18)+3(36)=0$

Thus,

$\left(\dfrac{u}{v}+\dfrac{v}{u}+\dfrac{v}{w}+\dfrac{w}{v}+\dfrac{u}{w}+\dfrac{w}{u}+3\right)(3^{u^2+v^2+w^2})^{u^3+v^3+w^3}=3(3^0)^0=3^1=3$
 

FAQ: How Do You Solve Complex Root Expressions in Polynomial Equations?

What is the purpose of finding the sum of expressions involving roots?

The purpose of finding the sum of expressions involving roots is to simplify and solve mathematical equations that contain square roots, cube roots, or other types of roots. This allows for easier manipulation and calculation of complex expressions.

What types of roots are typically involved in these types of expressions?

Expressions involving roots can contain any type of root, such as square roots (√), cube roots (∛), fourth roots (∜), and higher order roots. These roots can also be combined with other mathematical operations, such as addition, subtraction, multiplication, and division.

What is the general process for finding the sum of expressions involving roots?

The general process for finding the sum of expressions involving roots is to first simplify each individual root expression by factoring and simplifying the radicand (the number under the root symbol). Then, combine like terms and simplify the resulting expression. Finally, if needed, rationalize any remaining irrational roots by multiplying the numerator and denominator by the conjugate of the irrational root.

What are some common mistakes to avoid when finding the sum of expressions involving roots?

Common mistakes to avoid when finding the sum of expressions involving roots include forgetting to simplify the radicands, combining unlike terms, and not rationalizing irrational roots. It is also important to be careful when dealing with negative numbers and to always check your final answer for accuracy.

Are there any shortcuts or tricks for finding the sum of expressions involving roots?

Yes, there are some shortcuts and tricks that can be used to simplify the process of finding the sum of expressions involving roots. These include using the distributive property, memorizing the perfect squares and cubes, and recognizing patterns in the expressions. It is also helpful to practice and become familiar with common root identities.

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