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Rational expressions
Dynamic domain, simplification, operations and evaluation tasks with rational expressions.
Questions: 10Estimated time: 19 minutes
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Question 1out of 10
Simplify x² − 12²⁄x − 12, where x ≠ 12.
Factor the numerator and denominator. Cancel only shared nonzero factors, never individual terms in a sum. Keep the original denominator restrictions even when cancellation changes the denominator.
Question 2out of 10
Simplify x² − 12²⁄x − 12, where x ≠ 12.
Factor the numerator and denominator. Cancel only shared nonzero factors, never individual terms in a sum. Keep the original denominator restrictions even when cancellation changes the denominator.
Question 3out of 10
Add a/x and b/x when a = 2, b = 7.
With equal denominators, add numerators and keep the denominator: a/x + b/x = (a + b)/x. Require x ≠ 0.
Question 4out of 10
Evaluate (4x + 3)/x when x = 1.
Substitute the given nonzero x. Calculate the full numerator ax + b first, then divide by x. Parentheses mean the entire sum is divided by x.
Question 5out of 10
State the restrictions for 1/((x + 6)(x − 3)).
The denominator product must be nonzero, so neither factor may be zero. From (x − a)(x − b), exclude x = a and x = b.
Question 6out of 10
Simplify ((x2 − 64)/(x − 2)) · ((x − 2)/(x + 8)), where x ≠ 2 and x ≠ −8.
First recognise a difference of squares in the numerator:
x2 − a2 = (x − a)(x + a).
After substituting this product, the numerators and denominators contain matching factors. Cancel x − b and x + a. The remaining factor is the final answer.
Even after the factors are cancelled, the restrictions from the original expression remain: its denominators cannot equal zero.
Question 7out of 10
When may numerator and denominator factors of a rational expression be cancelled?
Factor the numerator and denominator. Cancel only shared nonzero factors, never individual terms in a sum. Keep the original denominator restrictions even when cancellation changes the denominator.
Question 8out of 10
Add a/x and b/x when a = 8, b = 7.
With equal denominators, add numerators and keep the denominator: a/x + b/x = (a + b)/x. Require x ≠ 0.
Question 9out of 10
State the restriction for 1/(x − 1).
A denominator cannot be zero. For denominator x − a, the excluded value is x = a. Thus x ≠ a, and the domain consists of all real numbers except a.
Question 10out of 10
Simplify ((x2 − 25)/(x − 1)) ÷ ((x − 5)/(x − 1)), where x ≠ 1 and x ≠ 5.
Replace division by a fraction with multiplication by its reciprocal:
A/B ÷ C/D = A/B · D/C.
Then factor the difference of squares:
x2 − a2 = (x − a)(x + a).
Cancel matching factors. Remember that the divisor cannot equal zero, so in addition to denominator restrictions you must exclude the value that makes the numerator of the second fraction zero.
Time left: 00:00:00
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Sample questions
- State the restriction for 1/(x − (___)).
- Simplify ___, where x ≠ ___.
- Add a/x and b/x when a = ___, b = ___.
- Simplify ((x2 − ___)/(x − ___)) · ((x − ___)/(x + ___)), where x ≠ ___ and x ≠ −___.
- Simplify ((x2 − ___)/(x − ___)) ÷ ((x − ___)/(x − ___)), where x ≠ ___ and x ≠ ___.
Rational expressions
A rational expression contains fractions involving polynomials. Its domain is restricted by denominators: no denominator may equal zero. Record excluded values before simplifying.
Operations with rational expressions
For multiplication, factor numerators and denominators and cancel common factors. For addition or subtraction, find a common denominator. Only factors may be cancelled, not separate terms in a sum; retain all restrictions from the original expression in the final result.
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