Rational Expressions Quiz
Questions: 16 · 10 minutes
1. Which expression is a rational expression, meaning it can be written as a quotient of two polynomials?
The exponential quotient 2ˣ/(x − 3)
The radical expression √x + 1
The polynomial quotient (x² − 4)/(x + 1)
The radical-denominator expression 1/(√x − 1)
2. What is the sum 2/x + 3/(x + 1)?
(5x + 2)/[x(x + 1)]
5/[x(x + 1)]
5/(2x + 1)
(5x + 3)/[x(x + 1)]
3. What is the least common denominator of 6/(x² − 4) and 5/[x(x − 2)]?
x²(x − 2)(x + 2), which repeats the factor x
x(x − 2), which omits the factor x + 2
x(x² − 4)(x − 2), which repeats the factor x − 2
x(x − 2)(x + 2), containing each distinct factor once
4. Solve 1/x + 1/(x + 2) = 3/[x(x + 2)].
x = 1
x = 3/2
x = 1/2
The equation has no permissible solution
5. For the original expression (x² − 9)/(x − 3), which domain restriction is required?
Exclude −3 because it makes the numerator zero
Permit every real value after simplifying the expression
Remove both −3 and 3 because they are numerator zeros
Exclude 3 because it makes the original denominator zero
6. Simplify 1/(x − 2) − 1/(x + 2).
Subtracting in the reverse order gives −4/(x² − 4)
The simplified difference is 4/(x² − 4)
Adding the cross-products gives 2x/(x² − 4)
Canceling the two from the numerator gives 2/(x² − 4)
7. For f(x) = (x² − 1)/(x² − 3x + 2), which equation gives the vertical asymptote?
The uncanceled denominator factor produces the vertical asymptote x = 2
The canceled factor at x = 1 produces a hole rather than an asymptote
The numerator zero at x = −1 gives an intercept, not a vertical asymptote
Equal polynomial degrees produce y = 1 as a horizontal asymptote
8. Divide and simplify [(x² − 4)/x] ÷ [(x + 2)/(3x)]. Which result includes all required restrictions?
Keeping the canceled factor leads to 3(x + 2), with 0 excluded
Failing to use the reciprocal gives (x − 2)/3, excluding 0 and −2
The quotient is 3(x − 2), with x ≠ 0 and x ≠ −2
Treating the divisor's numerator as x − 2 gives 3(x + 2), excluding 0 and 2
9. Evaluate (x² − 1)/(x + 1) at x = 2.
Direct substitution produces 3
The expression evaluates to 1
Taking the reciprocal gives 1/3
Combining the terms incorrectly gives 5/3
10. Simplify the complex fraction (1/x + 1/y) ÷ [1/(xy)], where x ≠ 0 and y ≠ 0.
The complex fraction simplifies to x + y
Inverting the wrong part produces xy/(x + y)
Combining denominators incorrectly produces 1/(x + y)
Ignoring the divisor leaves 1/x + 1/y unchanged
11. Simplify (x² − 9)/(x² − x − 6) and retain every restriction from the original expression.
Cancel the common factor to obtain (x + 3)/(x + 2), while retaining x ≠ 3 and x ≠ −2
Cancel the entire denominator and report the polynomial x + 3, with only x ≠ −2
Factor using opposite signs to obtain (x − 3)/(x − 2), excluding −3 and 2
Leave the original fraction unchanged and exclude only the numerator zeros −3 and 3
12. A workshop has a fixed cost of $500 and an additional cost of $20 per item. Its average cost per item is A(n) = (500 + 20n)/n. What is the average cost when 25 items are made?
An average cost of $20 per item
An average cost of $40 per item
An average cost of $520 per item
An average cost of $25 per item
13. A student solves (x + 1)/(x − 1) = 2/(x − 1) and obtains x = 1. What is the correct conclusion?
Accept 1 as the equation's unique solution
Treat every real number other than 1 as a solution
Substitute −1 as the unique solution instead
Reject 1 because it is excluded, leaving the equation with no solution
14. Solve x/(x − 2) = 3, taking the domain restriction into account.
A sign reversal suggests the incorrect value −3
The excluded denominator value 2 cannot be accepted
Solving x = 3(x − 2) produces the permissible answer x = 3
Using only the constant product leads incorrectly to 6
15. Multiply and simplify [2x/(x² − 9)] · [(x + 3)/4], retaining all restrictions from the original factors.
Cancel both binomial factors to get x/2, excluding only 3
The product is x/[2(x − 3)], with both −3 and 3 excluded
Using the wrong remaining factor gives x/[2(x + 3)], with −3 and 3 excluded
Multiplying without cancellation gives x(x + 3)/[2(x² − 9)], with no excluded values
16. The graph of g(x) = (x² − 4)/(x − 2) has a removable hole. What are its coordinates?
Using the other numerator zero suggests the incorrect location (−2, 0)
Reversing the input and output gives the incorrect point (4, 2)
Pairing the excluded input with a zero output gives the incorrect point (2, 0)
Evaluating the simplified rule x + 2 at the excluded input gives the hole (2, 4)