Drag the tiles to the correct boxes to complete the pairs. Match each rational expression to its simplest form.
[tex]\frac{2 m^2-4 m}{2(m-2)}[/tex]
[tex]\frac{m^2-2 m+1}{m-1}[/tex]
[tex]\frac{m^2-3 m+2}{m^2-m}[/tex]
[tex]\frac{m^2-m-2}{m^2-1}[/tex]
Answer (2)
Factor the numerator and denominator of each rational expression.
Cancel out any common factors in the numerator and denominator.
2 ( m − 2 ) 2 m 2 − 4 m = m
m − 1 m 2 − 2 m + 1 = m − 1
m 2 − m m 2 − 3 m + 2 = m m − 2
m 2 − 1 m 2 − m − 2 = m − 1 m − 2
Explanation
Problem Analysis We are given four rational expressions and we need to simplify each one and match it to its simplest form. The expressions are:
2 ( m − 2 ) 2 m 2 − 4 m
m − 1 m 2 − 2 m + 1
m 2 − m m 2 − 3 m + 2
m 2 − 1 m 2 − m − 2
Simplifying the Expressions Let's simplify each expression by factoring the numerator and the denominator and then canceling out common factors.
Expression 1: 2 ( m − 2 ) 2 m 2 − 4 m
Factor the numerator: 2 m 2 − 4 m = 2 m ( m − 2 ) .
So the expression becomes: 2 ( m − 2 ) 2 m ( m − 2 ) .
Cancel the common factor of 2 ( m − 2 ) : 2 ( m − 2 ) 2 m ( m − 2 ) = m .
Expression 2: m − 1 m 2 − 2 m + 1
Factor the numerator: m 2 − 2 m + 1 = ( m − 1 ) 2 = ( m − 1 ) ( m − 1 ) .
So the expression becomes: m − 1 ( m − 1 ) ( m − 1 ) .
Cancel the common factor of ( m − 1 ) : m − 1 ( m − 1 ) ( m − 1 ) = m − 1 .
Expression 3: m 2 − m m 2 − 3 m + 2
Factor the numerator: m 2 − 3 m + 2 = ( m − 1 ) ( m − 2 ) .
Factor the denominator: m 2 − m = m ( m − 1 ) .
So the expression becomes: m ( m − 1 ) ( m − 1 ) ( m − 2 ) .
Cancel the common factor of ( m − 1 ) : m ( m − 1 ) ( m − 1 ) ( m − 2 ) = m m − 2 .
Expression 4: m 2 − 1 m 2 − m − 2
Factor the numerator: m 2 − m − 2 = ( m − 2 ) ( m + 1 ) .
Factor the denominator: m 2 − 1 = ( m − 1 ) ( m + 1 ) .
So the expression becomes: ( m − 1 ) ( m + 1 ) ( m − 2 ) ( m + 1 ) .
Cancel the common factor of ( m + 1 ) : ( m − 1 ) ( m + 1 ) ( m − 2 ) ( m + 1 ) = m − 1 m − 2 .
Final Answer Therefore, the simplified forms of the given expressions are:
2 ( m − 2 ) 2 m 2 − 4 m = m
m − 1 m 2 − 2 m + 1 = m − 1
m 2 − m m 2 − 3 m + 2 = m m − 2
m 2 − 1 m 2 − m − 2 = m − 1 m − 2
Examples
Rational expressions are useful in many real-world applications, such as calculating the cost per item when buying in bulk. For example, if the total cost of buying x items is given by the expression x 2 + 5 x + 6 , and you want to find the cost per item, you would divide the total cost by the number of items, resulting in the rational expression x x 2 + 5 x + 6 . Simplifying this expression can help you easily determine the cost per item for different quantities purchased. Factoring and simplifying rational expressions is also used in physics, engineering, and economics to model various phenomena and solve problems.
The rational expressions were simplified by factoring and cancelling common factors. The final pairs are: 2 ( m − 2 ) 2 m 2 − 4 m = m , m − 1 m 2 − 2 m + 1 = m − 1 , m 2 − m m 2 − 3 m + 2 = m m − 2 , and m 2 − 1 m 2 − m − 2 = m − 1 m − 2 . This process showcases the importance of simplification in handling rational expressions.
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