Simplifying algebraic fractions - TickTockMaths - Free Printable

Problem: Simplify the algebraic fractions and find the odd one out in each row.

We will simplify each fraction step by step and identify which one in each row cannot be simplified further or behaves differently from the others.

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#### Row 1:
1. \( \frac{x+4}{x^2 + 5x + 4} \)
- Factor the denominator: \( x^2 + 5x + 4 = (x + 1)(x + 4) \)
- Simplify: \( \frac{x+4}{(x+1)(x+4)} = \frac{1}{x+1} \) (for \( x \neq -4 \))

2. \( \frac{x+5}{x^2 + 6x + 5} \)
- Factor the denominator: \( x^2 + 6x + 5 = (x + 1)(x + 5) \)
- Simplify: \( \frac{x+5}{(x+1)(x+5)} = \frac{1}{x+1} \) (for \( x \neq -5 \))

3. \( \frac{x+7}{x^2 + 9x + 14} \)
- Factor the denominator: \( x^2 + 9x + 14 = (x + 2)(x + 7) \)
- Simplify: \( \frac{x+7}{(x+2)(x+7)} = \frac{1}{x+2} \) (for \( x \neq -7 \))

4. \( \frac{x+7}{x^2 + 8x + 7} \)
- Factor the denominator: \( x^2 + 8x + 7 = (x + 1)(x + 7) \)
- Simplify: \( \frac{x+7}{(x+1)(x+7)} = \frac{1}{x+1} \) (for \( x \neq -7 \))

- Odd one out: The third fraction, \( \frac{x+7}{x^2 + 9x + 14} \), simplifies to \( \frac{1}{x+2} \), while the others simplify to \( \frac{1}{x+1} \).

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#### Row 2:
5. \( \frac{x+3}{x^2 + 6x + 9} \)
- Factor the denominator: \( x^2 + 6x + 9 = (x + 3)^2 \)
- Simplify: \( \frac{x+3}{(x+3)^2} = \frac{1}{x+3} \) (for \( x \neq -3 \))

6. \( \frac{x+6}{x^2 + 9x + 18} \)
- Factor the denominator: \( x^2 + 9x + 18 = (x + 3)(x + 6) \)
- Simplify: \( \frac{x+6}{(x+3)(x+6)} = \frac{1}{x+3} \) (for \( x \neq -6 \))

7. \( \frac{x+7}{x^2 + 10x + 21} \)
- Factor the denominator: \( x^2 + 10x + 21 = (x + 3)(x + 7) \)
- Simplify: \( \frac{x+7}{(x+3)(x+7)} = \frac{1}{x+3} \) (for \( x \neq -7 \))

8. \( \frac{x+4}{x^2 + 8x + 16} \)
- Factor the denominator: \( x^2 + 8x + 16 = (x + 4)^2 \)
- Simplify: \( \frac{x+4}{(x+4)^2} = \frac{1}{x+4} \) (for \( x \neq -4 \))

- Odd one out: The fourth fraction, \( \frac{x+4}{x^2 + 8x + 16} \), simplifies to \( \frac{1}{x+4} \), while the others simplify to \( \frac{1}{x+3} \).

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#### Row 3:
9. \( \frac{x+3}{x^2 - 2x - 15} \)
- Factor the denominator: \( x^2 - 2x - 15 = (x - 5)(x + 3) \)
- Simplify: \( \frac{x+3}{(x-5)(x+3)} = \frac{1}{x-5} \) (for \( x \neq -3 \))

10. \( \frac{x+7}{x^2 + 2x - 35} \)
- Factor the denominator: \( x^2 + 2x - 35 = (x - 5)(x + 7) \)
- Simplify: \( \frac{x+7}{(x-5)(x+7)} = \frac{1}{x-5} \) (for \( x \neq -7 \))

11. \( \frac{x+2}{x^2 - 3x - 10} \)
- Factor the denominator: \( x^2 - 3x - 10 = (x - 5)(x + 2) \)
- Simplify: \( \frac{x+2}{(x-5)(x+2)} = \frac{1}{x-5} \) (for \( x \neq -2 \))

12. \( \frac{x+3}{x^2 - 3x - 18} \)
- Factor the denominator: \( x^2 - 3x - 18 = (x - 6)(x + 3) \)
- Simplify: \( \frac{x+3}{(x-6)(x+3)} = \frac{1}{x-6} \) (for \( x \neq -3 \))

- Odd one out: The fourth fraction, \( \frac{x+3}{x^2 - 3x - 18} \), simplifies to \( \frac{1}{x-6} \), while the others simplify to \( \frac{1}{x-5} \).

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#### Row 4:
13. \( \frac{x^2 + 9x + 14}{x^2 + 10x + 21} \)
- Factor the numerator: \( x^2 + 9x + 14 = (x + 2)(x + 7) \)
- Factor the denominator: \( x^2 + 10x + 21 = (x + 3)(x + 7) \)
- Simplify: \( \frac{(x+2)(x+7)}{(x+3)(x+7)} = \frac{x+2}{x+3} \) (for \( x \neq -7 \))

14. \( \frac{x^2 + 10x + 16}{x^2 + 15x + 56} \)
- Factor the numerator: \( x^2 + 10x + 16 = (x + 2)(x + 8) \)
- Factor the denominator: \( x^2 + 15x + 56 = (x + 7)(x + 8) \)
- Simplify: \( \frac{(x+2)(x+8)}{(x+7)(x+8)} = \frac{x+2}{x+7} \) (for \( x \neq -8 \))

15. \( \frac{x^2 + 10x + 21}{x^2 + 11x + 24} \)
- Factor the numerator: \( x^2 + 10x + 21 = (x + 3)(x + 7) \)
- Factor the denominator: \( x^2 + 11x + 24 = (x + 3)(x + 8) \)
- Simplify: \( \frac{(x+3)(x+7)}{(x+3)(x+8)} = \frac{x+7}{x+8} \) (for \( x \neq -3 \))

16. \( \frac{x^2 + 13x + 42}{x^2 + 12x + 36} \)
- Factor the numerator: \( x^2 + 13x + 42 = (x + 6)(x + 7) \)
- Factor the denominator: \( x^2 + 12x + 36 = (x + 6)^2 \)
- Simplify: \( \frac{(x+6)(x+7)}{(x+6)^2} = \frac{x+7}{x+6} \) (for \( x \neq -6 \))

- Odd one out: All fractions in this row simplify to a different form, but none of them can be further simplified to a constant or a single term. However, the structure of the simplified forms is consistent, so there is no "odd one out" in the traditional sense. If we must choose, we could argue that the last fraction, \( \frac{x+7}{x+6} \), has a different denominator compared to the others.

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#### Row 5:
17. \( \frac{x^2 - 16x + 63}{x^2 - 11x + 18} \)
- Factor the numerator: \( x^2 - 16x + 63 = (x - 7)(x - 9) \)
- Factor the denominator: \( x^2 - 11x + 18 = (x - 2)(x - 9) \)
- Simplify: \( \frac{(x-7)(x-9)}{(x-2)(x-9)} = \frac{x-7}{x-2} \) (for \( x \neq 9 \))

18. \( \frac{x^2 + 4x - 12}{x^2 + 5x - 14} \)
- Factor the numerator: \( x^2 + 4x - 12 = (x - 2)(x + 6) \)
- Factor the denominator: \( x^2 + 5x - 14 = (x - 2)(x + 7) \)
- Simplify: \( \frac{(x-2)(x+6)}{(x-2)(x+7)} = \frac{x+6}{x+7} \) (for \( x \neq 2 \))

19. \( \frac{x^2 - 12x + 32}{x^2 - 3x - 40} \)
- Factor the numerator: \( x^2 - 12x + 32 = (x - 4)(x - 8) \)
- Factor the denominator: \( x^2 - 3x - 40 = (x - 8)(x + 5) \)
- Simplify: \( \frac{(x-4)(x-8)}{(x-8)(x+5)} = \frac{x-4}{x+5} \) (for \( x \neq 8 \))

20. \( \frac{x^2 - 2x - 35}{x^2 - 13x + 42} \)
- Factor the numerator: \( x^2 - 2x - 35 = (x - 7)(x + 5) \)
- Factor the denominator: \( x^2 - 13x + 42 = (x - 6)(x - 7) \)
- Simplify: \( \frac{(x-7)(x+5)}{(x-6)(x-7)} = \frac{x+5}{x-6} \) (for \( x \neq 7 \))

- Odd one out: All fractions in this row simplify to a different form, but none of them can be further simplified to a constant or a single term. However, if we look at the structure, the second fraction, \( \frac{x+6}{x+7} \), does not have a linear factor in the numerator that matches a linear factor in the denominator of the other fractions.

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Final Answer:

The odd ones out in each row are:
1. \( \boxed{3} \)
2. \( \boxed{8} \)
3. \( \boxed{12} \)
4. \( \boxed{16} \)
5. \( \boxed{18} \)