Questions: Introduction to algebraic fractions
Before attempting these questions, it is highly recommended that you read Guide: Introduction to algebraic fractions. You may also find reading Guide: Factorization useful.
Q1
For each algebraic fraction, state the restriction(s) on the denominator.
1.1. \(\displaystyle \quad \dfrac{3}{x}\)
1.2. \(\displaystyle \quad \dfrac{5}{x-4}\)
1.3. \(\displaystyle \quad -\dfrac{2}{2x+1}\)
1.4. \(\displaystyle \quad \dfrac{y+3}{y+5}\)
1.5. \(\displaystyle \quad \dfrac{4x}{x(3x-2)}\)
1.6. \(\displaystyle \quad \dfrac{t-1}{(t+2)(t-3)}\)
1.7. \(\displaystyle \quad \dfrac{5x+1}{x^2-16}\)
1.8. \(\displaystyle \quad \dfrac{2}{(x-1)^2}\)
1.9. \(\displaystyle \quad \dfrac{b}{b^2+4b}\)
1.10. \(\displaystyle \quad \dfrac{7}{(x+3)(2x-5)}\)
1.11. \(\displaystyle \quad -\dfrac{x-4}{x^2-5x+6}\)
1.12. \(\displaystyle \quad \dfrac{z+3}{z^2+2z-8}\)
Q2
In each of the following, find the missing value \(?\) that makes the two algebraic fractions equivalent.
2.1. \(\displaystyle \quad \dfrac{x}{3} = \dfrac{?}{9}\)
2.2. \(\displaystyle \quad \dfrac{2}{t} = \dfrac{10}{?}\)
2.3. \(\displaystyle \quad \dfrac{a+1}{4} = \dfrac{?}{12}\)
2.4. \(\displaystyle \quad \dfrac{3x}{5} = \dfrac{?}{20}\)
2.5. \(\displaystyle \quad \dfrac{5}{x+2} = \dfrac{?}{2(x+2)}\)
2.6. \(\displaystyle \quad \dfrac{t-3}{t} = \dfrac{?}{2t}\)
2.7. \(\displaystyle \quad \dfrac{4}{y-1} = \dfrac{12}{?}\)
2.8. \(\displaystyle \quad \dfrac{x}{x-4} = \dfrac{?}{3(x-4)}\)
2.9. \(\displaystyle \quad \dfrac{x+5}{2x} = \dfrac{3(x+5)}{?}\)
2.10. \(\displaystyle \quad \dfrac{2z-1}{z+3} = \dfrac{?}{3(z+3)}\)
2.11. \(\displaystyle \quad \dfrac{3}{x} = \dfrac{3(x+1)}{?}\)
2.12. \(\displaystyle \quad \dfrac{r}{r-2} = \dfrac{r(r+3)}{?}\)
2.13. \(\displaystyle \quad \dfrac{4x}{x+1} = \dfrac{?}{(x+1)^2}\)
2.14. \(\displaystyle \quad \dfrac{2x+1}{x-5} = \dfrac{6x+3}{?}\)
2.15. \(\displaystyle \quad \dfrac{5-z}{z+2} = \dfrac{?}{-2(z+2)}\)
Q3
Simplify each algebraic fraction.
3.1. \(\displaystyle \quad \dfrac{8y}{12y}\)
3.2. \(\displaystyle \quad \dfrac{5x^2}{15x}\)
3.3. \(\displaystyle \quad \dfrac{n^2+4n}{n}\)
3.4. \(\displaystyle \quad \dfrac{3x^2-9x}{6x}\)
3.5. \(\displaystyle \quad -\dfrac{4x}{10x}\)
3.6. \(\displaystyle \quad \dfrac{m^2-16}{m-4}\)
3.7. \(\displaystyle \quad \dfrac{x^2-1}{x+1}\)
3.8. \(\displaystyle \quad \dfrac{x^2+5x+6}{x+2}\)
3.9. \(\displaystyle \quad \dfrac{z^2+3z-10}{z-2}\)
3.10. \(\displaystyle \quad \dfrac{2x^2+7x+3}{x+3}\)
3.11. \(\displaystyle \quad \dfrac{y^2-10y+25}{y-5}\)
3.12. \(\displaystyle \quad \dfrac{4-x}{x-4}\)
3.13. \(\displaystyle \quad \dfrac{2x^2-8}{x^2-4}\)
3.14. \(\displaystyle \quad \dfrac{c^3-4c}{c^2-4}\)
3.15. \(\displaystyle \quad \dfrac{A^2-3A-10}{A^2-25}\)
After attempting the questions above, please click this link to find the answers.
Version history and licensing
v1.0: initial version created 12/25 by Donald Campbell as part of a University of St Andrews VIP project.