A student tries to simplify (x² + 5x)/(x + 5) by 'canceling the x' in both numerator and denominator, arriving at (x + 5)/1 = x + 5. What error did they make?
AThey should have divided numerator and denominator by 5, not x
BIn (x + 5), the x is being added, not multiplied — it is a term, not a factor, so it cannot be canceled. The correct step is to factor the numerator first: x(x + 5)/(x + 5) = x
CThe simplification is actually correct; x + 5 is the right answer
DThey should have used the quadratic formula to find values of x first
The cancellation rule applies only to factors — expressions that multiply the entire numerator or denominator. In (x + 5), the x is being added to 5; it does not multiply the whole expression. Writing (x² + 5x)/(x + 5) = (x + 5)/1 by crossing out x is the classic term-cancellation error. The correct procedure: factor the numerator as x(x + 5), then cancel the common factor (x + 5) to get x (with restriction x ≠ -5).
Question 2 Multiple Choice
After simplifying (x² − 9)/(x − 3) = (x + 3)(x − 3)/(x − 3) = x + 3, which statement is correct about the domain of the result?
AThe simplified form x + 3 is defined for all real numbers since there is no longer a denominator
BThe simplified form x + 3 must carry the restriction x ≠ 3, because the original expression was undefined at x = 3
CThe domain restriction only applies if we're plugging in specific values
DThe simplification changes the domain, so x = 3 is now a valid input
Simplification does not expand the domain. The original expression (x² − 9)/(x − 3) is undefined at x = 3 (division by zero). Even though the simplified form x + 3 would happily accept x = 3, the restriction must be carried forward — the two expressions are equivalent only on the domain where the original is defined. This is one of the most commonly forgotten steps in rational expression simplification.
Question 3 True / False
The expression (x + 7)/(x + 7) simplifies to 1 for most real values of x.
TTrue
FFalse
Answer: False
The expression equals 1 for all x ≠ −7. At x = −7, the denominator equals zero, making the expression undefined. Saying it equals 1 'for all real x' implicitly extends the domain beyond what the original expression allows. The correct simplified form is: 1, x ≠ −7.
Question 4 True / False
To simplify a rational expression correctly, you must factor the numerator and denominator completely before attempting to cancel anything.
TTrue
FFalse
Answer: True
Factoring first is the necessary prerequisite to canceling, because cancellation requires identifying common factors — things that multiply the entire numerator and denominator. Without factoring, expressions like (x² − 4)/(x + 2) appear to have no common parts, but after factoring the numerator as (x + 2)(x − 2), the common factor (x + 2) becomes visible. Attempting to cancel from unfactored expressions is the root cause of most rational expression errors.
Question 5 Short Answer
Explain why you can cancel (x − 2) in the expression (x − 2)(x + 3)/(x − 2) but cannot cancel the x in the expression (x + 3)/(x + 5), even though x appears in both numerator and denominator.
Think about your answer, then reveal below.
Model answer: In (x − 2)(x + 3)/(x − 2), the factor (x − 2) multiplies the entire numerator and is the entire denominator — it is a multiplicative factor of both. Canceling a factor means dividing both numerator and denominator by it, which is always valid (as long as it's not zero). In (x + 3)/(x + 5), the x in the numerator is part of a sum (x is being added to 3), not a factor multiplying the whole expression. Similarly in the denominator. There is no common multiplicative factor — x cannot be 'divided out' without changing the value of the expression for most x.
The fundamental rule: you can only cancel what divides the entire numerator AND the entire denominator. A term inside a sum or difference does not divide the whole sum. This is why 5/7 ≠ /7 — you can't cross out the 5 just because 5 appears somewhere.