Find the simplified quotient: [tex]$\frac{\frac{y^2+y}{y^2-2 y}}{4 y+4}$[/tex]
Answer (2)
Factor the numerator and denominator of the inner fraction: y 2 + y = y ( y + 1 ) and y 2 − 2 y = y ( y − 2 ) .
Factor the denominator of the main fraction: 4 y + 4 = 4 ( y + 1 ) .
Rewrite the main fraction as a division and multiply by the reciprocal: y ( y − 2 ) y ( y + 1 ) ⋅ 4 ( y + 1 ) 1 .
Cancel common factors to obtain the simplified quotient: 4 ( y − 2 ) 1 .
Explanation
Understanding the Problem We are given the expression 4 y + 4 y 2 − 2 y y 2 + y and asked to simplify it. This involves simplifying a complex fraction, which means we have a fraction in the numerator and a term in the denominator. Our goal is to combine these into a single, simplified fraction.
Factoring the Inner Fraction First, let's factor the numerator and denominator of the inner fraction:
Numerator: y 2 + y = y ( y + 1 ) Denominator: y 2 − 2 y = y ( y − 2 )
So, the inner fraction becomes y ( y − 2 ) y ( y + 1 ) .
Factoring the Denominator Now, let's factor the denominator of the main fraction:
4 y + 4 = 4 ( y + 1 )
Rewriting as Division We can rewrite the main fraction as a division:
4 ( y + 1 ) y ( y − 2 ) y ( y + 1 ) = y ( y − 2 ) y ( y + 1 ) ÷ 4 ( y + 1 )
Multiplying by the Reciprocal To divide by 4 ( y + 1 ) , we multiply by its reciprocal, which is 4 ( y + 1 ) 1 . So, we have:
y ( y − 2 ) y ( y + 1 ) ⋅ 4 ( y + 1 ) 1
Canceling Common Factors Now, we cancel common factors. We can cancel y and ( y + 1 ) from the numerator and denominator:
y ( y − 2 ) y ( y + 1 ) ⋅ 4 ( y + 1 ) 1 = ( y − 2 ) 1 ⋅ 4 1 = 4 ( y − 2 ) 1
Final Answer Therefore, the simplified quotient is 4 ( y − 2 ) 1 . We must also note the restrictions on y . Since we had y and y + 1 in the denominator at one point, y = 0 and y = − 1 . Also, y − 2 was in the denominator, so y = 2 .
Conclusion The simplified quotient is 4 ( y − 2 ) 1 .
Examples
Simplifying rational expressions is a fundamental skill in algebra and calculus. For instance, when analyzing the behavior of complex systems in physics or engineering, you might encounter rational functions that need simplification to understand their properties. Imagine you're designing a bridge and need to calculate the stress distribution. The equations might involve complex rational expressions, and simplifying them helps engineers identify critical points and ensure the bridge's stability. Similarly, in economics, simplifying rational functions can help analyze market trends and predict economic behavior more effectively.
The simplified quotient of the expression is 4 ( y − 2 ) 1 . Be sure to note that y = 0 , y = − 1 , and y = 2 to avoid any undefined expressions. These restrictions arise from the factors in the original expression.
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