Divide. Write your answer in lowest terms.

$\frac{y^2+2 y-3}{y^2-2 y+1} \div \frac{y+3}{y+1}$

To divide the rational expressions, we rewrite the division as multiplication, factor the resultant fractions, cancel common factors, and simplify. The final expression is y − 1 y + 1 ​ with restrictions on y: y  = 1 , − 3 , − 1 . Thus, we have obtained our answer in lowest terms.
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Answered by Anonymous | 2025-07-04

Rewrite the division as multiplication by the reciprocal: y 2 − 2 y + 1 y 2 + 2 y − 3 ​ × y + 3 y + 1 ​ .
Factor the numerator and denominator: ( y − 1 ) 2 ( y + 3 ) ( y − 1 ) ​ × y + 3 y + 1 ​ .
Cancel common factors: ( y − 1 ) 2 ​ ( y + 3 ) ​ ( y − 1 ) ​ ​ × ( y + 3 ) ​ y + 1 ​ .
Simplify the expression: y − 1 y + 1 ​ ​ .

Explanation

Problem Analysis We are asked to divide two rational expressions and simplify the result to its lowest terms. The given expression is:

Given Expression y 2 − 2 y + 1 y 2 + 2 y − 3 ​ ÷ y + 1 y + 3 ​

Dividing Rational Expressions To divide rational expressions, we multiply by the reciprocal of the second fraction. This gives us:

Rewriting as Multiplication y 2 − 2 y + 1 y 2 + 2 y − 3 ​ × y + 3 y + 1 ​

Factoring Quadratics Now, we factor the numerator and the denominator of the first fraction:

Factored Forms y 2 + 2 y − 3 = ( y + 3 ) ( y − 1 ) y 2 − 2 y + 1 = ( y − 1 ) ( y − 1 ) = ( y − 1 ) 2

Substituting Factors Substitute the factored forms into the expression:

Expression with Factors ( y − 1 ) 2 ( y + 3 ) ( y − 1 ) ​ × y + 3 y + 1 ​

Canceling Common Factors Now, we cancel the common factors from the numerator and the denominator. We can cancel ( y + 3 ) and one factor of ( y − 1 ) :

Simplified Expression ( y − 1 ) 2 ​ ( y + 3 ) ​ ( y − 1 ) ​ ​ × ( y + 3 ) ​ y + 1 ​ = y − 1 y + 1 ​

Restrictions on y So, the simplified expression is y − 1 y + 1 ​ . We must also consider the restrictions on y . The original expression had denominators y 2 − 2 y + 1 , y + 3 , and y + 1 . Therefore, y  = 1 , y  = − 3 , and y  = − 1 .

Final Answer Therefore, the final answer is:


Examples
Rational expressions are used in various fields, such as physics and engineering, to model relationships between different quantities. 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 expressions can appear in equations describing the motion of objects or the behavior of waves. Simplifying these expressions can help physicists understand and predict the behavior of physical systems.

Answered by GinnyAnswer | 2025-07-04