If a and b are roots of an equation, then find the equation with roots (omega)^a and (omega)^b.
Understand the Problem
The question is asking to find a new polynomial equation whose roots are expressed in terms of the original roots 'a' and 'b', specifically using omega raised to the power of those roots. This involves manipulating the original polynomial to reflect the transformation related to the new roots.
Answer
The new polynomial with roots $\omega^a$ and $\omega^b$ is: $$ Q(x) = x^2 - (\omega^a + \omega^b)x + \omega^{a+b} $$
Answer for screen readers
The new polynomial with roots $\omega^a$ and $\omega^b$ is:
$$ Q(x) = x^2 - (\omega^a + \omega^b)x + \omega^{a+b} $$
Steps to Solve
- Identify the Original Polynomial
Let's consider the original polynomial with roots $a$ and $b$. The polynomial can be expressed as:
$$ P(x) = (x - a)(x - b) $$
- Simplify the Original Polynomial
We can expand the original polynomial:
$$ P(x) = x^2 - (a + b)x + ab $$
Where $(a + b)$ is the sum of the roots and $ab$ is the product of the roots.
- Express New Roots as Functions of Original Roots
Next, we need to write the new roots as $\omega^a$ and $\omega^b$, where $\omega$ is a constant.
- Substitute New Roots into the Polynomial Form
This means we need to create a polynomial whose roots are $\omega^a$ and $\omega^b$.
We can rewrite the new polynomial as:
$$ Q(x) = (x - \omega^a)(x - \omega^b) $$
- Expand the New Polynomial
Expanding $Q(x)$ results in:
$$ Q(x) = x^2 - (\omega^a + \omega^b)x + \omega^a \omega^b $$
- Substitute the Values for New Roots
Substituting the values we have, we get:
$$ Q(x) = x^2 - (\omega^a + \omega^b)x + \omega^{a+b} $$
- Use the Sum and Product of the Original Roots
The expression for the roots translates into:
Sum of the new roots: $\omega^a + \omega^b$
Product of new roots: $\omega^{a + b}$
So the final polynomial can be simplified as:
$$ Q(x) = x^2 - (\omega^a + \omega^b)x + \omega^{a+b} $$
The new polynomial with roots $\omega^a$ and $\omega^b$ is:
$$ Q(x) = x^2 - (\omega^a + \omega^b)x + \omega^{a+b} $$
More Information
This form of the polynomial allows us to utilize transformations occurring with exponential forms of the roots. Expressing roots in this way can often be useful in complex analysis and series expansions where $e^{i\theta}$ transformations are utilized.
Tips
- Forgetting to properly expand the polynomial when changing roots could lead to incorrect coefficients.
- Confusing the new roots with their exponential forms without accounting for $a$ and $b$ leads to mistakes in the final expression.
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