Simplify: $\frac{1}{1-m+m^2}-\frac{1}{1+m+m^2}-\frac{2 m}{1-m^2+m^4}$
Answer (2)
Combine the first two fractions using a common denominator: 1 − m + m 2 1 − 1 + m + m 2 1 = 1 + m 2 + m 4 2 m .
Subtract the third term: 1 + m 2 + m 4 2 m − 1 − m 2 + m 4 2 m .
Find a common denominator and simplify: ( 1 + m 2 + m 4 ) ( 1 − m 2 + m 4 ) 2 m ( 1 − m 2 + m 4 ) − 2 m ( 1 + m 2 + m 4 ) = 1 + m 4 + m 8 − 4 m 3 .
The simplified expression is: 1 + m 4 + m 8 − 4 m 3 .
Explanation
Understanding the Problem We are given the expression 1 − m + m 2 1 − 1 + m + m 2 1 − 1 − m 2 + m 4 2 m to simplify. Our goal is to combine these fractions into a single, simplified expression.
Combining the First Two Fractions First, let's combine the first two fractions. To do this, we need a common denominator, which is ( 1 − m + m 2 ) ( 1 + m + m 2 ) . Multiplying these two expressions, we get: ( 1 − m + m 2 ) ( 1 + m + m 2 ) = 1 + m + m 2 − m − m 2 − m 3 + m 2 + m 3 + m 4 = 1 + m 2 + m 4 So, the common denominator is 1 + m 2 + m 4 .
Rewriting with Common Denominator Now, we can rewrite the first two fractions with the common denominator: 1 − m + m 2 1 − 1 + m + m 2 1 = ( 1 − m + m 2 ) ( 1 + m + m 2 ) ( 1 + m + m 2 ) − ( 1 − m + m 2 ) = 1 + m 2 + m 4 1 + m + m 2 − 1 + m − m 2 = 1 + m 2 + m 4 2 m
Subtracting the Third Term Next, we subtract the third term from the result: 1 + m 2 + m 4 2 m − 1 − m 2 + m 4 2 m To combine these fractions, we need a common denominator, which is ( 1 + m 2 + m 4 ) ( 1 − m 2 + m 4 ) . Multiplying these two expressions, we get: ( 1 + m 2 + m 4 ) ( 1 − m 2 + m 4 ) = 1 − m 2 + m 4 + m 2 − m 4 + m 6 + m 4 − m 6 + m 8 = 1 + m 4 + m 8 So, the common denominator is 1 + m 4 + m 8 .
Rewriting with Common Denominator Now, we can rewrite the expression with the common denominator: 1 + m 2 + m 4 2 m − 1 − m 2 + m 4 2 m = ( 1 + m 2 + m 4 ) ( 1 − m 2 + m 4 ) 2 m ( 1 − m 2 + m 4 ) − 2 m ( 1 + m 2 + m 4 ) = 1 + m 4 + m 8 2 m − 2 m 3 + 2 m 5 − 2 m − 2 m 3 − 2 m 5 = 1 + m 4 + m 8 − 4 m 3
Final Answer Therefore, the simplified expression is 1 + m 4 + m 8 − 4 m 3 .
Examples
Rational expressions are useful in various fields, such as physics and engineering, where they can model relationships between different variables. For example, in electrical engineering, rational functions can describe the impedance of a circuit as a function of frequency. Simplifying these expressions allows engineers to analyze and design circuits more efficiently. Similarly, in physics, rational functions can appear in the study of optics and wave mechanics, where they help describe the behavior of light and other waves.
We simplified the expression 1 − m + m 2 1 − 1 + m + m 2 1 − 1 − m 2 + m 4 2 m to arrive at 1 + m 4 + m 8 − 4 m 3 . The simplification involved finding common denominators and combining fractions. The final result is clear and concise and reflects the operations performed accurately.
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