## Simplifying Algebraic Fractions involving Quadratic Terms: Common Factors

As an experienced mathematics tutor, I have found that simplifying algebraic fractions involving quadratic terms can be a challenging task for many students. However, with a solid understanding of common factors and the right approach, anyone can master this essential skill. In this article, we will explore the process of simplifying algebraic fractions with quadratic terms, focusing on identifying and utilising common factors.

## Understanding Algebraic Fractions

Before we dive into the simplification process, let’s briefly review what algebraic fractions are. An algebraic fraction is a fraction in which the numerator and/or denominator contain algebraic expressions, such as polynomials. For example, $\displaystyle \frac{2x^2 + 6x + 4}{x + 1}$ is an algebraic fraction with a quadratic numerator and a linear denominator.

## Simplifying Algebraic Fractions with Common Factors

The key to simplifying algebraic fractions lies in identifying common factors in the numerator and denominator. By dividing both the numerator and denominator by their common factors, we can often simplify the fraction to its lowest terms.

### Step 1: Factorise the numerator and denominator

To begin, we must factorise the numerator and denominator of the algebraic fraction. This involves breaking down the quadratic expressions into their simplest factors.

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

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

Let’s consider the example mentioned earlier:

$\displaystyle \frac{2x^2 – 8}{3x^2 + 15x + 18}$

To factorise the numerator, we can use the difference of squares formula:

$\displaystyle 2x^2 – 8 = 2(x^2 – 4) = 2(x + 2)(x – 2)$

For the denominator, we can use the following steps:

$\displaystyle 3x^2 + 15x + 18 = 3(x^2 + 5x + 6) = 3(x + 2)(x + 3)$

### Step 2: Identify common factors

Once we have factorised the numerator and denominator, the next step is to identify any common factors between them. These common factors can be numbers, variables, or even more complex expressions.

In our example, we can see that there is a common factor of 2 between the numerator and denominator:

$\displaystyle \frac{2(x + 2)(x – 2)}{3(x + 2)(x + 3)}$

### Step 3: Divide out common factors

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

In our example, we can divide out the common factor of 2:

$\displaystyle \frac{2(x + 2)(x – 2)}{3(x + 2)(x + 3)} = \frac{2}{3} \times \frac{(x + 2)(x – 2)}{(x + 2)(x + 3)} = \frac{2}{3} \times \frac{x – 2}{x + 3}$

Thus, the simplified fraction is $\displaystyle \frac{2(x – 2)}{3(x + 3)}$.

## More Examples

Let’s work through a few more examples to reinforce the concept of simplifying algebraic fractions with common factors.

### Example 1

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

First, let’s factorise the numerator and denominator:

• Numerator: $\displaystyle 3x^2 + 6x = 3x(x + 2)$
• Denominator: $\displaystyle 9x = 3x \times 3$

Now, we can identify the common factor $3x$ and divide it out:

$\displaystyle \frac{3x(x + 2)}{3x \times 3} = \frac{x + 2}{3}$

### Example 2

$\displaystyle \frac{2x^2 – 18}{3x^2 + 21x + 36}$

Let’s factorise the numerator and denominator:

• Numerator: $\displaystyle 2x^2 – 18 = 2(x^2 – 9) = 2(x + 3)(x – 3)$
• Denominator: $\displaystyle 3x^2 + 21x + 36 = 3(x^2 + 7x + 12) = 3(x + 3)(x + 4)$

The common factor between the numerator and denominator is $(x + 3)$. Dividing it out, we get:

$\displaystyle \frac{2(x + 3)(x – 3)}{3(x + 3)(x + 4)} = \frac{2}{3} \times \frac{(x + 3)(x – 3)}{(x + 3)(x + 4)} = \frac{2}{3} \times \frac{x – 3}{x + 4}$

### Example 3

$\displaystyle \frac{4x^2 – 1}{2x – 1}$

Factorising the numerator and denominator:

• Numerator: $\displaystyle 4x^2 – 1 = (2x + 1)(2x – 1)$
• Denominator: $\displaystyle 2x – 1$

The common factor is $(2x – 1)$, so we divide it out:

$\displaystyle \frac{(2x + 1)(2x – 1)}{2x – 1} = 2x + 1$

## Practice Problems

To solidify your understanding of simplifying algebraic fractions with common factors, try solving these practice problems:

1. $\displaystyle \frac{x^2 – 25}{x – 5}$
2. $\displaystyle \frac{6x^2 + 18x}{12x}$
3. $\displaystyle \frac{x^2 + 7x + 12}{x^2 + 4x + 3}$
4. $\displaystyle \frac{9x^2 – 16}{3x + 4}$

Solutions:

1. $x + 5$
2. $\displaystyle \frac{x + 3}{2}$
3. $\displaystyle \frac{x + 4}{x + 1}$
4. $3x-4$

## Combining Common Factors with Other Techniques

In some cases, you may need to combine the common factor technique with other factorisation methods to simplify algebraic fractions fully. For example, you might encounter a fraction where the numerator and denominator share a common factor, but the remaining terms require factorisation using the difference of squares or sum/difference of cubes formulas.

Let’s consider an example:

$\displaystyle \frac{2x^3 – 16x}{x^2 – 9}$

In this case, both the numerator and denominator share a common factor of $x$. Let’s factorise them:

• Numerator: $\displaystyle 2x^3 – 16x = 2x(x^2 – 8)$
• Denominator: $\displaystyle x^2 – 9 = (x + 3)(x – 3)$

Now, we can divide out the common factor $x$:

$\displaystyle \frac{2x(x^2 – 8)}{x(x + 3)(x – 3)} = \frac{2(x^2 – 8)}{(x + 3)(x – 3)}$

To simplify further, we need to factorise $x^2 – 8$ using the difference of squares formula:

$\displaystyle x^2 – 8 = x^2 – (\sqrt{8})^2 = (x + 2\sqrt{2})(x – 2\sqrt{2})$

Substituting this back into our fraction:

$\displaystyle \frac{2(x + 2\sqrt{2})(x – 2\sqrt{2})}{(x + 3)(x – 3)}$

The fraction cannot be simplified further, so this is our final answer.

## Conclusion

Simplifying algebraic fractions with quadratic terms may seem intimidating at first, but by following the steps outlined in this article, you can break down the process into manageable parts. Remember to factorise the numerator and denominator, identify common factors, and divide them out to simplify the fraction. With practice and perseverance, you’ll soon be able to confidently tackle even the most complex algebraic fractions.

• 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.

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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