Simplify:
[tex]\begin{array}{l}
\frac{6 m}{m-3}+\frac{m}{m+1} \\
\frac{[?] m^2+\quad m}{(m-3)(m+\quad)}
\end{array}[/tex]
Answer (1)
Find a common denominator: ( m − 3 ) ( m + 1 ) .
Rewrite the expression with the common denominator: ( m − 3 ) ( m + 1 ) 6 m ( m + 1 ) + m ( m − 3 ) .
Expand and simplify the numerator: 6 m 2 + 6 m + m 2 − 3 m = 7 m 2 + 3 m .
The simplified expression is ( m − 3 ) ( m + 1 ) 7 m 2 + 3 m .
Explanation
Understanding the Problem We are given the expression m − 3 6 m + m + 1 m and we want to simplify it and express it in the form ( m − 3 ) ( m + ) [ ?] m 2 + m .
Finding a Common Denominator To add the two fractions, we need to find a common denominator. The common denominator is ( m − 3 ) ( m + 1 ) .
Rewriting with Common Denominator We rewrite the expression with the common denominator: m − 3 6 m + m + 1 m = ( m − 3 ) ( m + 1 ) 6 m ( m + 1 ) + ( m − 3 ) ( m + 1 ) m ( m − 3 )
Adding the Fractions Now, we add the fractions: ( m − 3 ) ( m + 1 ) 6 m ( m + 1 ) + m ( m − 3 )
Expanding the Numerator Next, we expand the numerator: 6 m ( m + 1 ) + m ( m − 3 ) = 6 m 2 + 6 m + m 2 − 3 m = 7 m 2 + 3 m
Simplified Expression So, the simplified expression is: ( m − 3 ) ( m + 1 ) 7 m 2 + 3 m
Identifying Missing Values Comparing this with the target form ( m − 3 ) ( m + ) [ ?] m 2 + m , we can identify the missing coefficients and constant. The coefficient of m 2 is 7, the coefficient of m is 3, and the constant in the denominator is 1.
Final Answer Thus, the simplified expression is ( m − 3 ) ( m + 1 ) 7 m 2 + 3 m .
Examples
This type of simplification is often used when solving equations involving rational expressions, such as in circuit analysis (electrical engineering), where you might need to combine impedance functions, or in chemical engineering, when dealing with reaction rates that are expressed as rational functions. Simplifying these expressions allows for easier manipulation and solution of the equations, leading to practical results in these fields.
