Simplifying algebraic fractions - TickTockMaths - Free Printable

Let’s work through each problem step by step.

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Question 3: Simplify the following algebraic fractions

We simplify by canceling common factors in numerator and denominator.

(a)
$\frac{(x + 6)(x + 3)}{(x + 3)}$
→ Cancel $(x + 3)$ top and bottom → $x + 6$

(b)
$\frac{(x - 1)(x + 1)}{(x - 1)}$
→ Cancel $(x - 1)$ → $x + 1$

(c)
$\frac{(x - 3)}{(x - 4)(x - 3)}$
→ Cancel $(x - 3)$ → $\frac{1}{x - 4}$

(d)
$\frac{(x + 7)^2}{(x + 7)} = \frac{(x + 7)(x + 7)}{(x + 7)}$
→ Cancel one $(x + 7)$ → $x + 7$

(e)
$\frac{(x - 3)(x + 2)}{(x + 2)(x + 9)}$
→ Cancel $(x + 2)$ → $\frac{x - 3}{x + 9}$

(f)
$\frac{(x + 2)(x + 4)^2}{(x + 4)} = \frac{(x + 2)(x + 4)(x + 4)}{(x + 4)}$
→ Cancel one $(x + 4)$ → $(x + 2)(x + 4)$

(g)
$\frac{(x + 1)(x + 2)(x + 3)}{(x + 2)(x + 3)(x + 4)}$
→ Cancel $(x + 2)$ and $(x + 3)$ → $\frac{x + 1}{x + 4}$

(h)
$\frac{x(x + 3)^2}{x(x + 1)(x + 3)} = \frac{x(x + 3)(x + 3)}{x(x + 1)(x + 3)}$
→ Cancel $x$ and one $(x + 3)$ → $\frac{x + 3}{x + 1}$

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Question 4: Simplify the following algebraic fractions

Here, we factorize quadratics first, then cancel common factors.

(a)
Numerator: $x^2 + 5x + 4 = (x + 1)(x + 4)$
Denominator: $x^2 + 4x + 3 = (x + 1)(x + 3)$
→ $\frac{(x + 1)(x + 4)}{(x + 1)(x + 3)}$ → Cancel $(x + 1)$ → $\frac{x + 4}{x + 3}$

(b)
Numerator: $x^2 + 6x + 9 = (x + 3)^2$
Denominator: $x^2 - 2x - 15 = (x - 5)(x + 3)$
→ $\frac{(x + 3)^2}{(x - 5)(x + 3)}$ → Cancel one $(x + 3)$ → $\frac{x + 3}{x - 5}$

(c)
Numerator: $x^2 - 2x = x(x - 2)$
Denominator: $x^2 + 2x - 8 = (x + 4)(x - 2)$
→ $\frac{x(x - 2)}{(x + 4)(x - 2)}$ → Cancel $(x - 2)$ → $\frac{x}{x + 4}$

(d)
Numerator: $x^2 - 7x + 10 = (x - 2)(x - 5)$
Denominator: $x^2 + 3x - 10 = (x + 5)(x - 2)$
→ $\frac{(x - 2)(x - 5)}{(x + 5)(x - 2)}$ → Cancel $(x - 2)$ → $\frac{x - 5}{x + 5}$

(e)
Numerator: $x^2 + 8x + 15 = (x + 3)(x + 5)$
Denominator: $x^2 - x - 12 = (x - 4)(x + 3)$
→ $\frac{(x + 3)(x + 5)}{(x - 4)(x + 3)}$ → Cancel $(x + 3)$ → $\frac{x + 5}{x - 4}$

(f)
Numerator: $x^2 + 13x + 40 = (x + 5)(x + 8)$
Denominator: $x^2 + 14x + 48 = (x + 6)(x + 8)$
→ $\frac{(x + 5)(x + 8)}{(x + 6)(x + 8)}$ → Cancel $(x + 8)$ → $\frac{x + 5}{x + 6}$

(g)
Numerator: $x^2 - 2x - 8 = (x - 4)(x + 2)$
Denominator: $x^2 + 6x - 40 = (x + 10)(x - 4)$
→ $\frac{(x - 4)(x + 2)}{(x + 10)(x - 4)}$ → Cancel $(x - 4)$ → $\frac{x + 2}{x + 10}$

(h)
Numerator: $x^2 + 10x + 24 = (x + 4)(x + 6)$
Denominator: $x^2 - 36 = (x - 6)(x + 6)$
→ $\frac{(x + 4)(x + 6)}{(x - 6)(x + 6)}$ → Cancel $(x + 6)$ → $\frac{x + 4}{x - 6}$

(i)
Numerator: $x^2 + 4x - 45 = (x + 9)(x - 5)$
Denominator: $x^2 + 10x + 9 = (x + 9)(x + 1)$
→ $\frac{(x + 9)(x - 5)}{(x + 9)(x + 1)}$ → Cancel $(x + 9)$ → $\frac{x - 5}{x + 1}$

(j)
Numerator: $x^2 + 11x = x(x + 11)$
Denominator: $x^2 - 121 = (x - 11)(x + 11)$
→ $\frac{x(x + 11)}{(x - 11)(x + 11)}$ → Cancel $(x + 11)$ → $\frac{x}{x - 11}$

(k)
Numerator: $x^2 - 1 = (x - 1)(x + 1)$
Denominator: $x^2 + x = x(x + 1)$
→ $\frac{(x - 1)(x + 1)}{x(x + 1)}$ → Cancel $(x + 1)$ → $\frac{x - 1}{x}$

(l)
Numerator: $x^2 - 15x + 44 = (x - 4)(x - 11)$
Denominator: $x^2 - 16 = (x - 4)(x + 4)$
→ $\frac{(x - 4)(x - 11)}{(x - 4)(x + 4)}$ → Cancel $(x - 4)$ → $\frac{x - 11}{x + 4}$

(m)
Numerator: $x^2 - x - 6 = (x - 3)(x + 2)$
Denominator: $x^2 - 2x - 3 = (x - 3)(x + 1)$
→ $\frac{(x - 3)(x + 2)}{(x - 3)(x + 1)}$ → Cancel $(x - 3)$ → $\frac{x + 2}{x + 1}$

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Final Answer:
Question 3:
(a) $x + 6$
(b) $x + 1$
(c) $\frac{1}{x - 4}$
(d) $x + 7$
(e) $\frac{x - 3}{x + 9}$
(f) $(x + 2)(x + 4)$
(g) $\frac{x + 1}{x + 4}$
(h) $\frac{x + 3}{x + 1}$

Question 4:
(a) $\frac{x + 4}{x + 3}$
(b) $\frac{x + 3}{x - 5}$
(c) $\frac{x}{x + 4}$
(d) $\frac{x - 5}{x + 5}$
(e) $\frac{x + 5}{x - 4}$
(f) $\frac{x + 5}{x + 6}$
(g) $\frac{x + 2}{x + 10}$
(h) $\frac{x + 4}{x - 6}$
(i) $\frac{x - 5}{x + 1}$
(j) $\frac{x}{x - 11}$
(k) $\frac{x - 1}{x}$
(l) $\frac{x - 11}{x + 4}$
(m) $\frac{x + 2}{x + 1}$