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Mastering Long Division and Rational Expressions in Algebra

Dive into algebra with a series of exercises focusing on long division, rational expression simplification, and equation solving. With problems designed for varying skill levels, you will learn to perform long division on polynomial expressions, simplify rational expressions by finding excluded values, and solve complex equations. This comprehensive practice will enhance your algebra skills and deepen your understanding of mathematical concepts. Ideal for students looking to improve their problem-solving techniques.

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Mastering Long Division and Rational Expressions in Algebra

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  1. Jeopardy

  2. 100 Divide (use long division) (45x4 – 60x2 + 30x) ÷ (15x)

  3. 200 Divide (use long division) (7x2 – 2x – 5) ÷ (x – 2)

  4. 300 Divide (use long division) (22 – 4x + 3x2) ÷ (x – 4)

  5. 400 Divide (use long division) (8x + x2 – 3) ÷ (2 – x)

  6. 500 Divide (use long division) (7 – x2) ÷ (x + 4)

  7. 100 Simplify the rational expression, if possible. Find the excluded values. 2x - 4 x2 + 8x – 20

  8. 200 Simplify the rational expression, if possible. Find the excluded values. 4x2 – 12x 2x2 – 5x – 3

  9. 300 Simplify the rational expression, if possible. Find the excluded values. x2 + 4x – 60 2x2 + 23x + 30

  10. 400 Simplify the rational expression, if possible. Find the excluded values. x - 4 x3 – 8x2 + 16x

  11. 500 Simplify the rational expression, if possible. Find the excluded values. x2 + 7x + 10 2x3 – 8x

  12. 100 Find the product or quotient. 3x2 – 10x – 8 ÷ 6x2 + x – 2 5x2 – 20x 30x2 – 120x

  13. 200 Find the product or quotient. 4x – 12 ● (2x2 + 11x – 40) x2 + 5x – 24 mmmmmmmm

  14. 300 Find the product or quotient. x2 + 2x – 35 ÷ 3x2 + 21x x2 – 3x – 10 9x + 18

  15. 400 Find the product or quotient. x2 + 8x + 15 ● x2 – 2x – 8 x2 + 7x + 10 3x2 + 9x

  16. 500 Find the product or quotient. x3 – x2 + 4x – 4 ÷ x2 + 7x – 8 10x3 5x2 + 40x

  17. 100 Find the sum or difference. 4x – x + 1 x2 – 1 x2 + 8x + 7

  18. 200 Find the sum or difference. x + 6 + x + 1 x2 – 4x – 12 x2 + 3x + 2

  19. 300 Find the sum or difference. x – 2 – x + 1 x2 – 6x + 9 x2 + 2x – 15

  20. 400 You paddle 8 miles upstream (against the current) and 8 miles downstream (with the current). The speed of the current is 1 mile per hour. • Write an equation that gives the total time t (in hours) as a function of your average speed r (in mph) in still water. • Find your total time if your average speed in still water is 3 mph.

  21. 500 You bike ride 50 miles from home. On your way back home, your average speed increases by 3 mph. • Write an equation that gives the total biking time t (in hours) as a function of your average speed r (in mph) when you are biking away from home. • Find your total time if you bike away from your home at an average speed of 15 mph.

  22. 100 Solve the equation. 2 – 6 = x x + 3 2x + 6 2

  23. 200 Solve the equation. x - 5x= -18 x – 4 x - 2 x - 2

  24. 300 Solve the equation. 1 – x – 1 = x (x + 1)2 x + 1

  25. 400 Solve the equation. x- 3x = -2x + 2 x – 2 x – 4 x2 – 6x + 8

  26. 500 Solve the equation. x + 7 – 1 = x + 10 x + 4 2x + 8

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