Algebraic Fractions Made Easy: Simplify Like a Pro 3

Simplifying Algebraic Fractions Common Factors

Algebraic Fractions Demystified: Simplify Common Factors by Quadratic Factorise

As an experienced mathematics tutor, I have encountered numerous students who struggle with simplifying algebraic fractions, particularly those involving quadratic terms. In this article, we will focus on simplifying algebraic fractions by finding common factors through quadratic factorisation. By the end of this guide, you will have the tools and knowledge needed to tackle these problems confidently.

Simplifying Algebraic Fractions with Quadratic Terms

Let’s learn how to simplify algebraic fractions containing quadratic terms by finding common factors.

Step 1: Factorise the numerator and denominator of each fraction

The first step in simplifying algebraic fractions is to factorise the numerator and denominator of each fraction. To do this, we need to find the factors of the quadratic terms.

For quadratic expressions, we can use various factorisation techniques such as:

  • Trial and error
  • Grouping
  • Decomposition
  • Sum or difference of cubes
  • Difference of squares

Let’s consider the first example:

$\displaystyle \frac{x^2+6x+9}{3x^2-27} \times \frac{x^2-6x+9}{x^2-3x}$

To factorise the numerators and denominators, we can use the decomposition method or recognise the perfect square trinomials:

$\displaystyle x^2+6x+9 = (x+3)^2$
$\displaystyle 3x^2-27 = 3(x^2-9) = 3(x+3)(x-3)$
$\displaystyle x^2-6x+9 = (x-3)^2$
$\displaystyle x^2-3x = x(x-3)$

Step 2: Identify common factors in each fraction

After factorising the numerators and denominators, we can identify any common factors within each fraction. These common factors can be numbers, variables, or even more complex expressions.

In our first example, we can see that:

  • In the first fraction, $(x+3)$ is a common factor in the numerator and denominator.
  • In the second fraction, $(x-3)$ is a common factor in the numerator and denominator.

$\require{cancel} \displaystyle \frac{\cancel{(x+3)}(x+3)}{3\cancel{(x+3)}(x-3)} \times \frac{\cancel{(x-3)}(x-3)}{x\cancel{(x-3)}}$

Step 3: Divide out common factors in each fraction

To simplify each algebraic fraction, we divide both the numerator and denominator by their common factors.

In our first example, we can divide out the common factor $(x+3)$ in the first fraction and $(x-3)$ in the second fraction:

$\displaystyle \frac{x+3}{3(x-3)} \times \frac{x-3}{x}$

Step 4: Multiply the simplified fractions

After simplifying each fraction, we can multiply the resulting fractions together.

In our first example, we get:

$\displaystyle \frac{x+3}{3(x-3)} \times \frac{x-3}{x} = \frac{(x+3)\cancel{(x-3)}}{3\cancel{(x-3)}x} = \frac{x+3}{3x}$

Thus, the simplified result is $\displaystyle \frac{x+3}{3x}$.

More Examples

Let’s work through the second example to reinforce the concept of simplifying algebraic fractions by finding common factors through quadratic factorisation.

Example 2

$\displaystyle \frac{4x^2+4x-24}{x^2+2x-8} \times \frac{x^2+x-12}{2x^2-8x-42}$

First, let’s factorise the numerators and denominators:

$\displaystyle 4x^2+4x-24 = 4(x^2+x-6) = 4(x+3)(x-2)$
$\displaystyle x^2+2x-8 = (x+4)(x-2)$
$\displaystyle x^2+x-12 = (x+4)(x-3)$
$\displaystyle 2x^2-8x-42 = 2(x^2-4x-21) = 2(x-7)(x+3)$

We can see that:

  • In the first fraction, $(x-2)$ is a common factor in the numerator and denominator.
  • In the second fraction, there are no common factors.

\( \begin{align} \require{cancel} \displaystyle \frac{4x^2+4x-24}{x^2+2x-8} \times \frac{x^2+x-12}{2x^2-8x-42} &= \frac{4(x+3)\cancel{(x-2)}}{(x+4)\cancel{(x-2)}} \times \frac{(x+4)(x-3)}{2(x-7)(x+3)} \\ &= \frac{4 \cancel{(x+3)}}{x+4} \times \frac{(x+4)(x-3)}{2(x-7)\cancel{(x+3)}} \\ &= \frac{4}{\cancel{x+4}} \times \frac{\cancel{(x+4)}(x-3)}{2(x-7)} \\ &= \cancel{4} \times \frac{x-3}{\cancel{2}(x-7)} \\ &= \frac{2(x-3)}{x-7} \end{align} \)

Thus, the simplified result is $\displaystyle \frac{2(x-3)}{x-7}$.

Practice Questions

To reinforce your understanding of simplifying algebraic fractions by finding common factors through quadratic factorisation, try solving these practice problems:

\( \begin{align} \displaystyle \frac{x^2+5x+6}{4x^2+4x-24} \times \frac{2x^2-8}{6x^2+18x+12} &= \frac{(x+2)(x+3)}{4(x^2+x-6)} \times \frac{2(x^2-4)}{6(x^2+3x+2)} \\ &= \frac{\cancel{(x+2)}(x+3)}{4(x+3)(x-2)} \times \frac{2(x+2)(x-2)}{6(x+1)\cancel{(x+2)}} \\ &= \frac{x+3}{4(x+3)(x-2)} \times \frac{\cancel{2}(x+2)(x-2)}{\cancel{6}(x+1)} \\ &= \frac{\cancel{x+3}}{4\cancel{(x+3)}(x-2)} \times \frac{(x+2)(x-2)}{3(x+1)} \\ &= \frac{1}{4\cancel{(x-2)}} \times \frac{(x+2)\cancel{(x-2)}}{3(x+1)} \\ &= \frac{1}{4} \times \frac{x+2}{3(x+1)} \\ &= \frac{x+2}{12(x+1)} \end{align} \)

\( \begin{align} \displaystyle \frac{9x^2-4}{3x^2-4x-4} \times \frac{3x^2-10x+8}{9x^2-18x+8} &= \frac{\cancel{(3x+2)}(3x-2)}{\cancel{(3x+2)}(x-2)} \times \frac{(3x-4)(x-2)}{(3x-2)(3x-4)} \\ &= \frac{3x-2}{x-2} \times \frac{\cancel{(3x-4)}(x-2)}{(3x-2)\cancel{(3x-4)}} \\ &= \frac{3x-2}{x-2} \times \frac{x-2}{3x-2} \\ &= 1 \end{align} \)

Conclusion

Simplifying algebraic fractions with quadratic terms may seem daunting at first, but by finding common factors through quadratic factorisation, it becomes much easier. Remember to factorise the numerators and denominators, identify common factors, and divide them out to simplify each fraction. Then, multiply the simplified fractions together to obtain the final result. By mastering these techniques, you’ll be able to confidently tackle a wide range of algebraic fraction problems involving quadratic terms.

Additional Tips and Tricks

  • When factorising quadratic expressions, always look for common factors first. This can save you time and simplify the process.
  • If you’re having trouble factorising a quadratic expression, try using the quadratic formula: $\displaystyle x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}$, where $a$, $b$, and $c$ are the coefficients of the quadratic expression in standard form $(ax^2 + bx + c)$.
  • Remember that if the numerator and denominator have no common factors, the fraction is already in its simplest form.
  • When simplifying algebraic fractions, be careful not to divide by zero. If you end up with a denominator that equals zero for a specific value of the variable, note that the fraction is undefined for that value.
  • Always keep an eye out for perfect square trinomials and the difference of squares, as they can often simplify the factorisation process and lead to more efficient simplification of the algebraic fractions.

By keeping these tips in mind and practising regularly, you’ll soon be able to simplify algebraic fractions with quadratic terms like a pro!

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