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Magazine Chapter 7: Problem 117
Consider the function g given by $$g(z)=\frac{z^{4}-z^{2}}{z-1}$$ Evaluate $$\frac{1}{w-w^{2}} \text { for } w=\frac{1-i}{10}$$
Short Answer
Expert verified
\[ \frac{10}{1 - \frac{4i}{5}} = \frac{250 + 200i}{41} \]
Step by step solution
01
- Simplify the function g(z)
The function given is: \[ g(z) = \frac{z^4 - z^2}{z - 1} \] Factor the numerator: \[ g(z) = \frac{z^2(z^2 - 1)}{z - 1} \] where \( z^2 - 1 \) can be written as \((z - 1)(z + 1)\). So, \[ g(z) = \frac{z^2 (z - 1)(z + 1)}{z - 1} \] Simplify by canceling \((z - 1)\): \[ g(z) = z^2(z + 1) \]
02
- Evaluate the expression for w
Given the expression \[ \frac{1}{w - w^2} \] and \( w = \frac{1 - i}{10} \). First, compute \( w^2 \): \[ w^2 = \left( \frac{1 - i}{10} \right)^2 = \frac{(1 - i)^2}{100} = \frac{1 - 2i + i^2}{100} = \frac{1 - 2i - 1}{100} = \frac{-2i}{100} = \frac{-i}{50} \] Now, find \( w - w^2 \): \[ w - w^2 = \frac{1 - i}{10} - \frac{-i}{50} = \frac{5(1 - i) + i}{50} = \frac{5 - 5i + i}{50} = \frac{5 - 4i}{50} = \frac{1 - 4i/5}{10} \] Finally, the value of the expression is: \[ \frac{1}{w - w^2} = \frac{10}{1 - \frac{4i}{5}} \]
03
- Rationalize the denominator
To simplify \( \frac{10}{1 - \frac{4i}{5}} \), multiply both the numerator and the denominator by the conjugate of the denominator: \[ \frac{10}{1 - \frac{4i}{5}} \times \frac{1 + \frac{4i}{5}}{1 + \frac{4i}{5}} = \frac{10(1 + \frac{4i}{5})}{1 - (\frac{4i}{5})^2} = \frac{10(1 + \frac{4i}{5})}{1 - \frac{16i^2}{25}} = \frac{10(1 + \frac{4i}{5})}{1 + \frac{16}{25}} = \frac{10(1 + \frac{4i}{5})}{\frac{41}{25}} = \frac{250(1 + \frac{4i}{5})}{41} = \frac{250 + 200i}{41} \]
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Polynomial Factorization
Polynomial factorization is the process of breaking down a polynomial into simpler polynomials that, when multiplied together, give you the original polynomial. In our given function, \( g(z) = \frac{z^4 - z^2}{z - 1} \), we need to factor the numerator. The polynomial in the numerator can be written as \( z^2(z^2 - 1) \). Since \( z^2 - 1 \) can be factored further to \( (z - 1)(z + 1) \), the fraction simplifies to:
\( g(z) = \frac{z^2(z-1)(z+1)}{z-1} \). By canceling \((z-1)\) from the numerator and denominator, we get the simplified function:
\( g(z) = z^2(z + 1) \).
Understanding polynomial factorization is crucial because it allows us to simplify complex expressions and solve polynomial equations more easily.
\( g(z) = \frac{z^2(z-1)(z+1)}{z-1} \). By canceling \((z-1)\) from the numerator and denominator, we get the simplified function:
\( g(z) = z^2(z + 1) \).
Understanding polynomial factorization is crucial because it allows us to simplify complex expressions and solve polynomial equations more easily.
Complex Numbers
Complex numbers are numbers that have both a real part and an imaginary part. They are usually written in the form \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit with the property \(i^2 = -1\). In the given exercise, we are provided with \( w = \frac{1 - i}{10} \).
To evaluate \( \frac{1}{w - w^2} \), we first calculate \( w^2 \). Use the given form of \(w\):
To evaluate \( \frac{1}{w - w^2} \), we first calculate \( w^2 \). Use the given form of \(w\):
- Calculate \( w^2 = \left( \frac{1 - i}{10} \right)^2 \), which simplifies to \( \frac{-i}{50} \).
- Next, find \( w - w^2 \): \( \frac{1 - i}{10} - \frac{-i}{50} = \frac{5(1 - i) + i}{50} \), which simplifies to \( \frac{5 - 4i}{50} = \frac{1 - \frac{4i}{5}}{10} \).
Rationalizing Denominators
Rationalizing the denominator involves removing any complex numbers or radicals from the denominator of a fraction. This process often involves multiplying the numerator and the denominator by the conjugate of the denominator.
In our problem, after simplifying \( \frac{1}{ w - w^2} \), we found we needed to rationalize \( \frac{10}{1 - \frac{4i}{5}} \). The conjugate of \( 1 - \frac{4i}{5} \) is \( 1 + \frac{4i}{5} \). We multiply the fraction by:
\( \frac{1 + \frac{4i}{5}}{1 + \frac{4i}{5}} \). By doing this, the expression becomes:
Rationalizing denominators is particularly useful because it makes complex expressions more manageable and easier to interpret.
In our problem, after simplifying \( \frac{1}{ w - w^2} \), we found we needed to rationalize \( \frac{10}{1 - \frac{4i}{5}} \). The conjugate of \( 1 - \frac{4i}{5} \) is \( 1 + \frac{4i}{5} \). We multiply the fraction by:
\( \frac{1 + \frac{4i}{5}}{1 + \frac{4i}{5}} \). By doing this, the expression becomes:
- Numerator: \( 10(1 + \frac{4i}{5}) \)
- Denominator: \( 1 + \frac{16}{25} = \frac{41}{25} \)
Rationalizing denominators is particularly useful because it makes complex expressions more manageable and easier to interpret.
Function Evaluation
Function evaluation involves substituting a given value or expression into a function and simplifying the result. With the function \( g(z) = z^2(z + 1) \) and another expression \( \frac{1}{ w - w^2} \), we need to substitute the complex number \( w = \frac{1 - i}{10} \). Following these steps:
Function evaluation is vital as it gives us the actual output of a function when an input is provided, allowing us to understand and use mathematical models.
- Calculate \( w \).
- Calculate additional required values such as \( w^2 \).
- Substitute these into the given functions or expressions.
Function evaluation is vital as it gives us the actual output of a function when an input is provided, allowing us to understand and use mathematical models.
