Subtract the rational expressions:
3y+2

y2−2y+1
−
2y+3

y2−2y+1

Simplify if possible.

Question

Subtract the rational expressions:  
3y+2

y2−2y+1
−
2y+3

y2−2y+1

Simplify if possible.

BlackTom AI · Accepted Answer

To tackle this subtraction, I start by recognizing that both fractions share the same denominator, which makes the process straightforward.
First, write the subtraction with a common denominator: (3y+2)/(y^2-2y+1) − (2y+3)/(y^2-2y+1) = [(3y+2) − (2y+3)] / (y^2−2y+1).
Next, simplify the numerator: (3y+2) − (2y+3) = 3y + 2 − 2y − 3 = y − 1.
So the expression becomes (y − 1) / (y^2 − 2y + 1).
Now factor the denominator: y^2 − 2y + 1 = (y − 1)^2. Substituting gives (y − 1) / (y − 1)^2.
At this point one factor of (y − 1) cancels between numerator and denominator, leaving 1 / (y − 1), provided y ≠ 1 to avoid division by zero.
It’s important to state the domain restriction: since the original denominators include y^2 − 2y + 1 = (y − 1)^2, we must exclude y = 1 from the domain to keep expressions defined.
If you consider alternative paths, such as attempting to cancel before combining the fractions, you’d still need to ensure you’re only canceling common factors that are not zero; otherwise you’d run into undefined expressions. In this problem, canceling (y − 1) is valid after the combined numerator is found, but not if y = 1.
Thus, the simplified result is the expression 1/(y − 1) with the domain restriction y ≠ 1.

ALGII141-027 5.4 Practice Problems

Subtract the rational expressions: 3y+2 y2−2y+1 − 2y+3 y2−2y+1 Simplify if possible.

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