Rational Functions: Background You’ll Need 3

  • Simplify fractions using factors

simplifying fractions

Simplifying fractions involves finding common factors in the numerator and denominator, then canceling them out.

From your work with basic fractions, you might remember that [latex]\frac{12}{18} = \frac{2 \cdot 6}{3 \cdot 6} = \frac{2}{3}[/latex] by canceling the common factor of 6. The same principle applies to algebraic expressions.

Simplify [latex]\frac{(x-3)(x+2)}{(x-3)(x+5)}[/latex]

[latex]\begin{aligned}= \frac{(x-3)(x+2)}{(x-3)(x+5)} \quad \text{identify common factor } (x-3)\\ &= \frac{x+2}{x+5} \quad \text{cancel the common factor} \end{aligned}[/latex]

Most rational expressions won’t be in factored form so it’s important we review how to factor as well.

There are multiple factoring techniques:

  • GCF:

    Find the GCF of [latex]12x^3y^2[/latex] and [latex]18x^2y^4[/latex]

    [latex]\begin{aligned} 12x^3y^2 &= 2^2 \cdot 3 \cdot x^3 \cdot y^2\ 18x^2y^4 \\&= 2 \cdot 3^2 \cdot x^2 \cdot y^4\ \text{GCF} \\&= 2 \cdot 3 \cdot x^2 \cdot y^2 = 6x^2y^2 \end{aligned}[/latex]

  • Trinomial:

    Factor [latex]x^2 - 9x + 18[/latex]

    [latex]= (x - 3)(x - 6) \quad \text{find two numbers that multiply to 18 and add to -9} \end{aligned}[/latex]

  • Difference of squares:

    Factor [latex]4x^2 - 25[/latex]

    [latex]\begin{aligned} 4x^2 - 25 = \& (2x)^2 - 5^2 \quad \text{recognize difference of squares}\\ &= (2x + 5)(2x - 5) \end{aligned}[/latex]

Simplify [latex]\frac{x^2 - 4}{x^2 + 4x + 4}[/latex]

[latex]\begin{aligned} \frac{x^2 - 4}{x^2 + 4x + 4} & = \frac{(x-2)(x+2)}{(x+2)^2} \quad \text{factor both numerator and denominator} \\ &= \frac{(x-2)(x+2)}{(x+2)(x+2)} \quad \text{identify common factor}\\ &= \frac{x-2}{x+2} \quad \text{cancel one factor of } (x+2) \end{aligned}[/latex]

You can only cancel common factors, never common terms. For instance, in [latex]\frac{x + 3}{x + 5}[/latex], you cannot cancel the [latex]x[/latex] terms because they are being added, not multiplied.

Now you’re ready to work with more complex rational expressions and understand how rational functions behave.